Abstract

We briefly survey the theory of thermodynamic formalism for uniformly
hyperbolic systems, and then describe several recent approaches to the
problem of extending this theory to non-uniform hyperbolicity. The first of
these approaches involves Markov models such as Young towers,
countable-state Markov shifts, and inducing schemes. The other two are less
fully developed but have seen significant progress in the last few years: these
involve coarse-graining techniques (expansivity and specification) and
geometric arguments involving push-forward of densities on admissible
manifolds.

Keywords

Mathematics Subject Classification

Primary 37D25, 37D35

1. Introduction

1.1. The General Setting

Thermodynamic formalism, i.e., the formalism of equilibrium
statistical physics, was adapted to the study of dynamical systems in the
classical works of [Ruelle1972], [Ruelle1978], [Sinai1968], [Sinai1972], and [Bowen1970], [Bowen1974], [Bowen2008]. It provides an ample collection of
methods for constructing invariant measures with strong statistical properties.
In particular, this includes constructing a certain
"physical"
measure known as the SRB measure
(for Sinai-Ruelle-Bowen).

The general ideas can be given as follows. Let $ (X,d)$ be a
compact metric space and $ f \colon X \to X$ a continuous map of finite topological
entropy. Fix a continuous function $ \ph\colon X\to \RR$, which we will refer to as a
potential
. Denote by $ \MMM(f)$ the space of all $ f$-invariant Borel
probability measures on X. Given $ \mu\in \MMM(f)$, the free energy
of the system with respect to $ \mu$ is \begin{eqnarray*} E_\mu(\ph) := -\left(h_\mu(f) + \int_X \ph\,d\mu\right), \end{eqnarray*} where $ h_\mu(f)$ is
the Kolmogorov-Sinai (measure-theoretic) entropy of $ (X,f,\mu)$.
Optimizing over all invariant measures gives the
topological pressure
\begin{eqnarray*} P(\ph) := -\inf_{\mu\in \MMM(f)} E_\mu(\ph) = \sup_{\mu\in \MMM(f)} \left(h_\mu(f) + \int_X \ph\,d\mu\right), \end{eqnarray*} and a measure achieving this extremum is called an equilibrium measure
(or equilibrium state
). Note that it suffices to take the infimum (supremum) over the space
$ \MMM^e(f) \subset \MMM(f)$ of ergodic
measures.

The variational principle
relates the definition of pressure as an extremum over invariant measures to an
alternate definition in terms of growth rates. Given $ \eps> 0$ and
$ n\in \NN$, a set $ E\subset X$ is $ (n,\eps)$-
separated
if points in $ E$ can be distinguished at a scale $ \eps$ within
$ n$ iterates; more precisely, if for every $ x,y\in E$ with
$ x\ne y$, there is $ 0\le k\le n$ such that $ d(f^kx,f^ky)\ge\eps$. Then one has

The sum in
(1.1)
is a partition sum
that quantifies "weighted orbit complexity at spatial scale $ \eps$ and
time scale $ n$"; $ P(\ph)$ represents the growth rate of this
complexity as time increases. In the particular case $ \ph=0$, the value
$ P(0)$ is the topological entropy $ h_{\text{top}}(f)$ of the map $ f$.

Thermodynamic formalism is most useful when the system possesses
some degree of hyperbolic behavior, so that orbit complexity increases
exponentially. The most complete results are available when $ f$ is
uniformly hyperbolic; we discuss these in Sect. 1.2. In this article
we focus on non-uniformly hyperbolic systems, and we discuss the general
picture in Sect. 1.3. Our emphasis will be on general techniques
rather than on specific examples. In particular, we discuss Markov models
(including Young towers) in Sects. 2–4,
coarse-graining techniques (based on expansivity and specification) in Sect.
5, and push-forward (geometric) approaches
(based on newly introduced standard pairs approach) in Sect. 6.

1.2. Uniformly Hyperbolic Maps (Sinai, Ruelle,
Bowen)

1.2.1. General Thermodynamic
Results

We refer the reader to ([Katok and Hasselblatt1995], [Brin and Stuck2002]) for fundamentals of uniform
hyperbolicity theory and to ([Bowen2008], [Parry and Pollicott1990]) for a complete
description of thermodynamic formalism for uniformly hyperbolic systems.
Consider a compact smooth Riemannian manifold $ M$ and a
$ C^1$ diffeomorphism $ f\colon M\to M$. A compact invariant set
$ \Lambda\subset M$ is called hyperbolic
if for every $ x\in\Lambda$ the tangent space $ T_xM$ admits an invariant
splitting $ T_xM=E^s(x)\oplus E^u(x)$ into stable
and unstable
subspaces with uniform contraction and expansion: this means that there are
numbers $ c> 0$ and $ 0< \lambda< 1$ such that for every $ x\in\Lambda$:

One can show that the subspaces $ E^s$ and $ E^u$ depend
Hölder continuously on $ x$; in particular, there is
$ k> 0$ such that $ \angle(E^s(x),E^u(x))\ge k$ for every $ x\in\Lambda$.

Moving from the tangent bundle to the manifold itself, for every
$ x\in\Lambda$ one can construct local stable
$ V^s(x)$ and unstable
$ V^u(x)$ manifolds
(also called leaves
) through $ x$ which are tangent to $ E^s(x)$ and $ E^u(x)$
respectively and depend Hölder continuously on $ x$ ([Katok and Hasselblatt1995], Sect. 6.2). In
particular, there is $ \varepsilon> 0$ such that for any $ x,y\in\Lambda$ for which
$ d(x,y)\le\varepsilon$ one has that the intersection $ V^s(x)\cap V^u(y)$ consists of a single
point (here $ d(x,y)$ denotes the distance between points $ x$ and
$ y$ induced by the Riemannian metric on $ M$). We denote
this point by $ [x,y]$.

A hyperbolic set $ \Lambda$ is called
locally maximal
if there is a neighborhood $ U$ of $ \Lambda$ such that for any
invariant set $ \Lambda'\subset U$ we have that $ \Lambda'\subset\Lambda$. In other words,
$ \Lambda=\bigcap_{n\in\mathbb{Z}}\,f^n(U)$. One can show that a hyperbolic set $ \Lambda$ is locally
maximal if and only if for any $ x,y\in\Lambda$ which are sufficiently close to each
other, the point $ [x,y]$ lies in $ \Lambda$ ([Katok and Hasselblatt1995], Sect. 6.4).

Given a locally maximal hyperbolic set and a Hölder
continuous potential function, thermodynamic formalism produces unique
equilibrium measures with strong ergodic properties: before stating the
theorem we recall some notions from ergodic theory for the reader's
convenience. Let $ (X,\mu)$ be a Lebesgue space with a probability measure
$ \mu$ and $ T\colon X\to X$ an invertible measurable transformation that
preserves $ \mu$.

(1) The Bernoulli property.
Let $ Y$ be a finite set and $ \nu$ a probability measure on
$ Y$ (that is, a probability vector). One can associate to $ (Y,\nu)$
the two-sided Bernoulli shift $ \sigma\colon Y^{\mathbb{Z}}\to Y^{\mathbb{Z}}$ defined by $ (\sigma y)_n=y_{n+1}$,
$ n\in\mathbb{Z}$; this preserves the measure $ \kappa$ given as the direct
product of $ \ZZ$ copies of $ \nu$. We say that $ (T,\mu)$ is a
Bernoulli automorphism (or "has the Bernoulli property") if $ (T,\mu)$ is
metrically isomorphic to the Bernoulli shift $ (\sigma,\kappa)$ associated to some
Lebesgue space $ (Y,\nu)$ and we also say that $ \mu$ is a
Bernoulli measure. More generally, one can take $ (Y,\nu)$ to be a
Lebesgue space, so $ \nu$ is metrically isomorphic to Lebesgue
measure on an interval together with at most countably many atoms. For all
the cases we discuss, it suffices to take $ Y$ finite. ×1

exponential decay of correlations
(EDC)
with respect to $ \mathcal{H}$ if there is $ 0< \theta< 1$ satisfying: for every
$ h_1, h_2\in\mathcal{H}$ there is $ K=K(h_1,h_2)> 0$ such that for every $ n> 0$
\begin{eqnarray*} \text{Cor}_n(h_1,h_2)\le K \theta^{n}; \end{eqnarray*}

polynomial decay of
correlations (PDC)
with respect to $ \mathcal{H}$ if there is $ \alpha> 0$ satisfying: for every
$ h_1, h_2\in\mathcal{H}$ there is $ K=K(h_1,h_2)> 0$ such that for every $ n> 0$
\begin{eqnarray*} \text{Cor}_n(h_1,h_2)\le Kn^\alpha. \end{eqnarray*}

(3) The Central Limit Theorem.
Say that a measurable function $ h$ is cohomologous to a constant if
there is a measurable function $ g$ and a constant $ c$
such that $ h=g\circ T - g + c$ almost everywhere. We say that the transformation
$ T$ satisfies the Central Limit Theorem (CLT) for functions in a
class $ \mathcal{H}$ if for any $ h\in\mathcal{H}$ that is not cohomologous to a
constant, there exists $ \gamma> 0$ such that \begin{eqnarray*} \mu\left\{x : \frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\left(h(T^i(x))-\int h\,d\mu\right)< t\right\}\rightarrow\frac{1}{\gamma\sqrt{2\pi}}\int_{-\infty}^t e^{-\tau^2/2\gamma^2}\,d\tau. \end{eqnarray*}

Before stating the formal result, we point out that uniformly
hyperbolic systems (and many non-uniformly hyperbolic ones) satisfy various
other statistical properties, which we do not discuss in detail in this survey.
These include large deviations principles ([Orey and Pelikan1988], [Young1990], [Kifer1990], [Pfister and Sullivan2005], [Melbourne and Nicol2008], [Rey-Bellet and Young2008], [Climenhaga et al.2013]), Borel–Cantelli
lemmas ([Chernov and Kleinbock2001], [Dolgopyat2004], [Kim2007], [Gouëzel2007], [Gupta et al.2010], [Haydn et al.2013]), the almost sure invariant
principle ([Denker and Philipp1984], [Melbourne and Nicol2005], [Melbourne and Nicol2009]), and many more
besides.

Theorem 1.1.

Let $ \Lambda$ be a locally maximal
hyperbolic set for $ f$, and assume that $ f|\Lambda$ is
topologically transitive. This means that there is a point $ x\in\Lambda$ whose
trajectory is everywhere dense, i.e., $ \Lambda=\overline{\{f^nx : n\in \ZZ\}}$. An equivalent definition is
that for any two non-empty open sets $ U$ and $ V$ there
is $ n\in\mathbb{Z}$ such that $ f^n(U)\cap V\ne\emptyset$. ×2 Then for any Hölder
continuous potential $ \ph$, the following are true:

(1) Existence:
there is an equilibrium measure $ \mu_\ph$.

(2) Uniqueness:
$ \mu_\ph$ is the only equilibrium measure for $ \ph$.

(3) Ergodic and statistical properties:

(a) the Bernoulli
property: there is $ A\subset\Lambda$ and $ n> 0$ such that the sets
$ f^k(A)$, $ 0\le k< n$ are (essentially) disjoint and cover
$ \Lambda$, $ f^n(A)=A$, and $ (f^n|A,\mu_\ph)$ has the Bernoulli property;

(b) exponential
decay of correlations: there are $ A,n$ as above such that
$ (f^n|A,\mu_\ph)$ has EDC with respect to the class of Hölder continuous
functions.

(c) the
Central Limit Theorem: $ \mu_\ph$ satisfies the CLT with respect to the
class of Hölder continuous functions.

The proof of Theorem 1.1 uses
the fact that $ f|\Lambda$ can be represented by a subshift of finite type
via a Markov partition
. Recall that a $ p\times p$ transition matrix That is, a matrix whose
entries $ a_{ij}$ are each equal to 0 or 1. ×3 $ A$ determines a
subshift of finite type (SFT) $ (\Sigma_A,\sigma)$ as the (left) shift $ \sigma(\omega)_i=\omega_{i+1}$ on
the space $ \Sigma_A$ of two-sided infinite sequences $ \omega=(\omega_i) \in \{1,\dots, p\}^\ZZ$ which
are admissible with respect to $ A$; that is, for which $ a_{\omega_i\omega_{i+1}}=1$
for all $ i\in \ZZ$.

Recall also that a finite partition $ \mathcal{R}=\{R_1,\dots,R_p\}$ of $ \Lambda$ is a
Markov partition if the following are true.

(2) $ R_i=\overline{\text{int}\,R_i}$
Here $ \text{int}\,R_i$ means the interior of the set $ R_i$ in the
relative topology. ×4 and for any $ 1\le i,j\le p$,
$ i\ne j$ we have that $ \text{int}\,R_i\cap\text{int}\,R_j=\emptyset$; this guarantees that the coding
map is injective away from the boundaries.

(3) Each set $ R_i$ is a rectangle
, i.e., for any $ x,y\in R_i$ we have that $ z=[x,y]\in R_i$; this is the local product structure
(or hyperbolic product structure
) of the partition elements.

The first construction of Markov partitions was obtained by [Adler and Weiss1967], [Adler and Weiss1970], and independently by [Berg1967], in the particular case of hyperbolic
automorphisms of the 2-torus. They observed that the map allowed a symbolic
representation by a subshift of finite type and that this can be used to study
its ergodic properties. Sinai realized that existence of Markov partitions is a
rather general phenomenon and he constructed Markov partitions for general
Anosov diffeomorphisms, see [Sinai1968]. Furthermore, in [Sinai1972] he observed the analogy between the
symbolic models of Anosov diffeomorphisms and lattice gas models in
physics—the starting point in developing the thermodynamic
formalism. Finally, in the more general setting of locally maximal hyperbolic
sets Markov partitions were constructed by [Bowen1970].

Markov partitions allow one to obtain a symbolic representation of
the map $ f|\Lambda$ by subshifts of finite type. More precisely, let
$ \mathcal{R}=\{R_1,\dots,R_p\}$ be a finite Markov partition of $ \Lambda$. Consider the
subshift of finite type $ (\Sigma_A,\sigma)$ with the transition matrix $ A$
whose entries are given by $ a_{ij}=1$ if $ f(\mathrm{int}\, R_i)\cap \mathrm{int}\, R_j\ne\emptyset$ and $ a_{ij}=0$
otherwise. One can show that for every $ \omega=(\omega_i)\in\Sigma_A$ the intersection
$ \bigcap_{i\in\mathbb{Z}}f^{-i}(R_{\omega_i}) $ is not empty and consists of a single point $ \pi(\omega)$. This
defines the coding map
$ \pi\colon\Sigma_A\to\Lambda$, which is characterized by the fact that $ f^i(\pi(\omega)) \in R_{\omega_i}$ for all
$ i\in \ZZ$ (thus $ \omega$ "codes" the orbit of $ \pi(\omega)$).

Proposition 1.2.

The map $ \pi$ has the following
properties:

(1) $ \pi$
is Hölder continuous;

(2) $ \pi$ is a conjugacy between the shift
$ \sigma$ and the map $ f|\Lambda$, i.e., $ (f|\Lambda)\circ\pi=\pi\circ\sigma$;

(3) $ \pi$ is
one-to-one on the set $ \Sigma'\subset\Sigma$ which consists of points $ \omega$ for
which the trajectory of the point $ \pi(\omega)$ never hits the boundary of the
Markov partition.

Consider a Hölder continuous potential
$ \varphi$ on $ \Lambda$. By Proposition 1.2, the function
$ \tilde\varphi$ on $ \Sigma_A$ given by $ \tilde\varphi(\omega)=\varphi(\pi(\omega))$ is Hölder
continuous. Thus in order to prove Theorem 1.1 it
suffices to study thermodynamic formalism for Hölder continuous
potentials for SFTs. The starting point for this theory is the following result
of [Parry1964], which uses Perron–Frobenius
theory to deal with the case $ \ph=0$. The corresponding equilibrium
measure is the measure of maximal entropy (MME) for which $ h_\mu(f)=h_{\text{top}}(f)$.

Theorem 1.3.

Let $ A$ be a transition matrix such
that $ A^n> 0$ for some $ n\in \NN$, and let $ \Sigma_A$ be the
corresponding SFT.

(1) The
topological entropy of $ \Sigma_A$ is $ \log\lambda$, where $ \lambda> 1$ is
the maximal eigenvalue of $ A$ guaranteed by the
Perron–Frobenius theorem.

Theorem 1.3 was adapted to non-zero potentials by [Ruelle1968], [Ruelle1976], replacing the transition matrix with a
transfer operator
. Ruelle's version of the Perron–Frobenius theorem for this transfer
operator is at the heart of the classical results in thermodynamic formalism for
SFTs, and hence, for uniformly hyperbolic systems. Roughly speaking the idea
is the following.

(1) Replace the
two-sided SFT $ \Sigma_A$ with its one-sided version $ \Sigma_A^+$, and
define the transfer operator associated to $ \ph$ on $ C(\Sigma_A^+)$ by
It is
instructive to consider the case $ \ph=0$ and write down the action of
$ \LLL_0$ on the (finite-dimensional) space of functions constant on
1-cylinders, where the action is given by the (transpose of the) transition
matrix $ A$. ×5 \begin{eqnarray*} (\LLL_\ph f)(x) = \sum_{\sigma y=x} e^{\ph(y)} f(y). \end{eqnarray*}

(2) Show that $ \LLL_\ph$ has a
largest eigenvalue $ \lambda$ and that the rest of the spectrum lies inside a
disc with radius $ {< }\lambda$ (the spectral gap
property).

(3)
Instead of the left and right eigenvalues $ h$ and $ v$, find
a positive eigenfunction $ h\in C(\Sigma_A^+)$ for $ \LLL_\ph$, and an eigenmeasure
$ \nu\in \MMM(\Sigma_A^+)$ for the dual $ \LLL_\ph^*$.

(4) Obtain the unique equilibrium state as
$ d\mu = h\,d\nu$.

We stress that this result (and hence Theorem 1.1) may
not hold if the the potential function fails to be Hölder continuous, see
[Hofbauer1977], [Sarig2001a], [Pesin and Zhang2006].

[]

Figure 1 The pressure function for
a
typical hyperbolic sets; b
a hyperbolic attractor; c
a non-uniformly hyperbolic map with a phase transition.

1.2.2. Thermodynamic Formalism for the
Geometric $ t$-Potential

Returning from SFTs to the setting of uniformly hyperbolic smooth
systems, the most significant potential function is the geometric
$ t$ -potential
: a family of potential functions $ \ph_t(x):= -t\log \abs{\Jac(df|E^u(x))}$ for $ t\in\RR$. Since the
subspaces $ E^u(x)$ depend Hölder continuously on $ x$,
the potential $ \ph_t$ is Hölder continuous for each $ t$
whenever $ f$ is $ C^{1+\alpha}$; in particular, it admits a unique
equilibrium measure $ \mu_t$. Furthermore, the pressure function
$ P(t) := P(\ph_t)$ is well defined for all $ t$, is convex, decreasing, and
real analytic in $ t$, as in Fig. 1a.

There are certain values of $ t$ that are particularly
important.

When $ t=0$, we obtain the topological entropy
$ \htop(f)$ as $ P(0)$, and the unique MME as $ \mu_0$.

Since $ P$ is strictly decreasing and has
$ P(0) > 0$ and $ P(t)\to -\infty$ as $ t\to\infty$, there is a unique number
$ t_0> 0$ for which $ P(t_0)=0$. The equation $ P(t)=0$ is called
Bowen's equation
. In the two-dimensional case its root is the Hausdorff dimension of
$ \Lambda\cap V^u(x)$ The value of the Hausdorff dimension does not depend
on $ x$. ×6 and the equilibrium measure
$ \mu_{t_0}$ achieves this Hausdorff dimension (i.e., is the measure of
maximal dimension) ([Bowen1979], [Ruelle1982], [McCluskey and Manning1983]).

To further study the properties of the pressure function (and
$ t_0$ in particular) we recall the notion of the Lyapunov exponent.
Given $ x\in\Lambda$ and $ v\in T_xM$, define the Lyapunov exponent
\begin{eqnarray*} \chi(x,v)=\limsup_{n\to\infty}\,\frac1n\log\|df^nv\|. \end{eqnarray*} For every $ x\in\Lambda$ the function $ \chi(x,\cdot)$ takes on
finitely many values $ \chi_1(x)\le\cdots\le\chi_d(x)$ where $ d=\dim M$. The functions
$ \chi_i(x)$ are Borel and are invariant under $ f$; in particular, if
$ \mu$ is an ergodic measure, then $ \chi_i(x)=\chi_i(\mu)$ is constant almost
everywhere for each $ i=1,\dots, d$, and the numbers $ \chi_i(\mu)$ are called
the Lyapunov exponent of the measure
$ \mu$. If none of these numbers is equal to zero, $ \mu$ is
called a hyperbolic measure
; It
is assumed that some of these numbers are positive while others are negative.
×7 note that when $ \Lambda$ is a
hyperbolic set for $ f$, every invariant measure supported on
$ \Lambda$ is hyperbolic. The Margulis–Ruelle inequality (see
[Ruelle1979, Barreira and Pesin2013]) says that

and in particular implies that $ t_0\le 1$, since the sum in
(1.3)
is equal to $ -\int \ph_1\,d\mu$, and hence $ h_\mu(f)+\int\ph_1\,d\mu \le 0$ for every ergodic
$ \mu$.

1.2.3. Hyperbolic Attractors

We consider the particular case when $ \Lambda$ is a topological attractor
for $ f$. This means that there is a neighborhood $ U\supset\Lambda$
such that $ \overline{f(U)}\subset U$ and $ \Lambda=\bigcap_{n\ge 0}f^n(U)$. It is not difficult to see that for
every $ x\in\Lambda$, the local unstable manifold $ V^u(x)$ is contained in
$ \Lambda$; Indeed, for any $ y\in V^u(x)$ the trajectory of
$ y$, $ \{f^n(y)\}_{n\in\mathbb{Z}}$ lies in $ U$ and hence, must belong to
$ \Lambda$ since it is locally maximal. ×8 the same is true for the global
unstable manifold through $ x$. Therefore, the attractor contains all
the global unstable manifolds of its points. On the other hand the intersection
of $ \Lambda$ with stable manifolds of its points is usually a Cantor set.

In the case when $ \Lambda$ is a hyperbolic attractor we have that
$ t_0=1$ (see [Bowen2008]), so $ P(t)$ is as in Fig. 1b. The equilibrium state $ \mu_1$ is a
hyperbolic ergodic measure for which the Margulis–Ruelle inequality
(1.3)
becomes equality. By [Ledrappier and Young1985], this implies that
$ \mu_1$ has absolutely continuous conditional measures along unstable
manifolds; that is, there is a collection $ \RRR$ of local unstable
manifolds $ V^u$ and a measure $ \eta$ on $ \RRR$ such
that $ \mu_1$ can be written as

where the measures $ \mu_{V^u}$ are absolutely continuous with
respect to the leaf volumes $ m_{V^u}$. A hyperbolic measure
$ \mu$ satisfying
(1.4)
is said to be a Sinai–Ruelle–Bowen
(SRB) measure, and it can be shown that such measures are physical
: the set of generic
points \begin{eqnarray*} G_\mu := \left\{x\in M \mid \frac 1n \sum_{k=0}^{n-1} \varphi(f^k(x)) \to \int\ph\,d\mu \text{ for all continuous } \ph\colon M\to \RR \right\} \end{eqnarray*} has positive volume, and so $ \mu$ is the
appropriate invariant measure for studying "physically relevant" trajectories.
The discussion above shows that when $ \Lambda$ is a hyperbolic attractor,
SRB measures are precisely the equilibrium states for the geometric potential
$ \ph_1$.

1.3. Non-uniformly Hyperbolic
Maps.

1.3.1. Definition of Non-uniform
Hyperbolicity

A $ C^{1+\alpha}$ diffeomorphism $ f$ of a compact smooth
Riemannian manifold $ M$ is
non-uniformly hyperbolic
on an invariant Borel subset $ S\subset M$ if there are a measurable
$ df$-invariant decomposition of the tangent space $ T_xM=E^s(x)\oplus E^u(x)$ for
every $ x\in S$ and measurable $ f$-invariant functions
$ \varepsilon(x)> 0$ and $ 0< \lambda(x)< 1$ such that for every $ 0< \varepsilon\le\varepsilon(x)$ one can
find measurable functions $ c(x)> 0$ and $ k(x)> 0$ satisfying for every
$ x\in S$:

The last property means that the estimates in (1) and (2) can deteriorate but
with sub-exponential rate.

If $ \mu$ is an invariant measure for $ f$ with
$ \mu(S)=1$, then by the Multiplicative Ergodic theorem, if for almost every
$ x\in S$ the Lyapunov exponents at $ x$ are all nonzero, i.e.,
$ \mu$ is a hyperbolic measure, then $ f$ is non-uniformly
hyperbolic on $ S$.

1.3.2. Possibility of Phase
Transitions and Non-hyperbolic Behavior.

A general theory of thermodynamic formalism for non-uniformly
hyperbolic maps is far from being complete, although certain examples here
are well-understood. They include one-dimensional maps, where the pressure
function $ P(t)=P(\ph_t)$ associated with the family of geometric potentials may
behave as in the uniformly hyperbolic case, or may exhibit new phenomena
such as phase transitions
(points of non-differentiability where there is more than one equilibrium
measure). The latter is illustrated in Fig. 1c
and is most thoroughly studied for the Manneville–Pomeau map
$ x\mapsto x+x^{1+\alpha} \pmod 1$, where $ \alpha\in (0,1)$ controls the degree of intermittency
at the neutral fixed point. In this example one has the following behavior
([Pianigiani1980], [Thaler1980], [Thaler1983], [Lopes1993], [Pollicott and Weiss1999], [Liverani et al.1999], [Young1999], [Sarig2002], [Hu2004]).

Hyperbolic behavior for
$ t< 1$: the pressure function $ P(t)$ is real analytic and
decreasing on $ (-\infty,1)$, and for every $ t$ in this range, the
geometric $ t$-potential $ \ph_t$ has a unique equilibrium
measure $ \mu_t$, which is Bernoulli, has EDC, and satisfies the CLT
with respect to the class of Hölder continuous potentials. In a nutshell,
for $ t\in (-\infty,1)$, the thermodynamics of this system is just as in the case of
uniform hyperbolicity.

Phase
transition at
$ t=1$: the pressure function $ P(t)$ is non-differentiable at
$ t=1$, and $ \ph_1$ has two ergodic equilibrium measures. One
of these is the absolutely continuous invariant
probability measure
$ \mu_1$ (which plays the role of SRB measure), and the other is the
point mass $ \delta_0$ on the neutral fixed point. For $ \alpha\in(0,1)$ the
measure $ \mu_1$ is finite but for $ \alpha\ge 1$, a new phenomenon
occurs: the intermittent behavior becomes strong enough that while there is
still an absolutely continuous invariant measure, it is infinite. At the same
time, the pressure function for $ \alpha\ge 1$ becomes differentiable at
$ t=1$, and the measure $ \delta_0$ becomes the unique equilibrium
measure. ×9 The measure $ \mu_1$ is
Bernoulli and decay of correlations is polynomial
(in particular, subexponential).

Non-hyperbolic behavior for
$ t> 1$: for every $ t\in (1,\infty)$, the unique equilibrium state for
$ \ph_t$ is the point mass $ \delta_0$, which has zero entropy and
zero Lyapunov exponent.

Similar results for the geometric $ t$-potential are available for other
classes of one-dimensional maps (e.g., unimodal and multimodal maps) and
rather specific higher-dimensional examples (e.g., polynomial and rational
maps and (piecewise) non-uniformly expanding maps); in some of these
examples phase transitions occur while others are without phase transitions.
As a small sample of the recent literature on the topic, we mention only
([Bruin and Keller1998], [Makarov and Smirnov2000], [Oliveira2003], [Alves et al.2005], [Przytycki and Rivera-Letelier2007], [Pesin and Senti2008], [Bruin and Todd2008], [Bruin and Todd2009], [Dobbs2009], [Iommi and Todd2010], [Przytycki and Rivera-Letelier2011], [Li and Rivera-Letelier2014a], [Li and Rivera-Letelier2014b]), as well as the
comprehensive and far-reaching discussion of thermodynamics for interval
maps with critical points in [Dobbs and Todd2015].

Our goal in the rest of this paper is not to discuss these results, which
rely on the specific structure of the examples being studied (or on the absence
of a contracting direction); rather, we want to discuss the recently developed
techniques for studying multi-dimensional non-uniformly hyperbolic systems,
with particular emphasis on recent results that have the potential to be applied
very generally, although they do not yet give as complete a picture as the one
outlined above. These general results have been obtained in the last few years
and represent an actively evolving area of research.

1.3.3. Different Types of
Equilibrium Measures.

Before describing the general methods, we recall some basic notions
from non-uniform hyperbolicity; see [Barreira and Pesin2007] for more complete
definitions and properties. Let $ M$ be a compact smooth manifold
and $ f\colon M\to M$ a $ C^{1+\alpha}$ diffeomorphism. Recall that a point
$ x\in M$ is called Lyapunov–Perron
regular
if for any basis $ \{v_1,\dots, v_p\}$ of $ T_xM$, \begin{eqnarray*} \liminf_{n\to\pm\infty}\frac1n\log V(n)= \limsup_{n\to\pm\infty}\frac1n\log V(n) =\sum_{i=1}^p\,\chi_i(x,v_i), \end{eqnarray*} where
$ V(n)$ is the volume of the parallelepiped built on the vectors
$ \{df^nv_1,\dots, df^nv_p\}$.

Let $ \mathcal{R}$ be the set of all Lyapunov–Perron regular
points. The Multiplicative Ergodic theorem claims that this set has full measure
with respect to any invariant measure. Consider now the set $ \Gamma\subset\mathcal{R}$ of
points for which all Lyapunov exponents are nonzero, and let $ \MMM^e(f,\Gamma)\subset \MMM^e(f)$ be
the set of all ergodic measures that give full weight to the set $ \Gamma$;
these are hyperbolic measures and they form the class of measures where it is
reasonable to attempt to recover some of the theory of uniformly hyperbolic
systems.

Let $ \ph$ be a measurable potential function; note that we
cannot a priori assume more than measurability if we wish to include the
family of geometric potentials, since in general the unstable subspace varies
discontinuously and so $ \ph_t$ is not a continuous function. On the
other hand, for surface diffeomorphisms [Sarig2013] constructed Markov partitions with
countably many partition elements (see Sect. 3 below), and
showed [Sarig2011] that the function $ \ph_t$ can be
lifted to a function on the symbolic space that is globally well-defined and is
Hölder continuous. This can be used to study equilibrium measures for
this function. ×10 Consider the hyperbolic pressure
defined by using only hyperbolic measures:

Say that $ \mu_\ph$ is a hyperbolic
equilibrium measure
if $ -E_{\mu_\ph}(\ph)=P_\Gamma(\ph)$. For the Manneville–Pomeau example above, we have
$ P_\Gamma(\ph_t)=P(\ph_t)$ for every $ t\in\RR$, and the equilibrium measure
$ \mu_t$ is the unique hyperbolic equilibrium measure for every
$ t\le 1$, Note that for $ t=1$ it is no longer the unique
equilibrium measure, but it is the only hyperbolic one. ×11 while for $ t> 1$ there is
no hyperbolic equilibrium measure, since $ \delta_0$ has zero Lyapunov
exponent.

One could also fix a threshold $ h> 0$ and consider the set
$ \MMM^e(f,\Gamma,h)$ of all measures in $ \MMM^e(f,\Gamma)$ whose entropies are greater
than $ h$; restricting our attention to measures from this class gives
the restricted pressure Because $ \MMM^e(f,\Gamma,h)$ is not compact, the existence of
an optimizing measure in
(1.6)
becomes a more subtle issue. Although it may happen that the value of
$ P_\Gamma^h(\ph)$ is achieved by a measure $ \mu$ whose entropy may not
be greater than $ h$, the restriction to measures in the class
$ \MMM^e(f,\Gamma,h)$ is often made to ensure a certain "liftability" condition, which
may still be satisfied by $ \mu$; see Theorem 2.3 and the discussion
in that section. ×12

For the Manneville–Pomeau example, we have for every
$ t\in \RR$, This is reminiscent of the use of Katok horseshoes to
approximate (with respect to entropy) an arbitrary system with a uniformly
hyperbolic one [Katok1980], which was recently generalized to
pressure by [Sánchez-Salas2015]. ×13 \begin{eqnarray*} \lim_{h\to 0} P_\Gamma^h(\ph_t) = P_\Gamma(\ph_t) = P(\ph_t). \end{eqnarray*} In addition to the use
of $ \mu_t^h$ to approximate non-hyperbolic measures by hyperbolic ones,
the above approach is also useful when one can identify a (not necessarily
invariant) subset $ \mathcal{A}\subset X$ of "bad"
points away from which the dynamics exhibits good hyperbolic behavior; then
putting $ h> h_{\text{top}}(f,\mathcal{A})$ guarantees that we consider only measures to which
$ \mathcal{A}$ is invisible. Note that since $ \mathcal{A}$ is not
assumed to be invariant, one should use the definition of the topological
entropy based on the Carathéodory construction of dimension-like
characteristics for dynamical systems ([Bowen1973], [Pesin1997]). ×14 This concept originated in the
work of Buzzi on piecewise invertible continuous maps of compact metric
spaces [Buzzi1999], An important goal there was to study the
notion of $ h$ -isomorphism
, which asks for two systems to have (measure-theoretically) conjugate
subsystems that carry all ergodic measures with large enough entropy, even if
the whole systems are not conjugate. ×15 but it is reasonable to consider it
in other situations. For example, if the set $ \mathcal{A}$ is an elliptic
island and the potential function is sufficiently large on $ \mathcal{A}$, then
the equilibrium measure may be a zero entropy measure sitting outside the set
with non-zero Lyapunov exponents. Putting any positive threshold removes
this measure from consideration. ×16

One could also impose a threshold in other ways. For example, one
could fix a reference potential $ \psi$ and a threshold $ p < P(\psi)$,
then restrict attention to the set $ \MMM^e(f,\Gamma,\psi,p)$ of all measures in $ \MMM^e(f,\Gamma)$
for which $ -E_\mu(\psi)> p$. Optimizing $ E_\mu(\ph)$ over this restricted set of
measures gives another notion of thresholded equilibrium states that may be
useful; again, it is often natural to take $ p=P_S(\ph)$ as the topological pressure
of $ f$ on a (not necessarily invariant) subset $ S\subset M$ of bad
points. Another approach would be to consider only measures whose Lyapunov
exponents are sufficiently large; it may be that this is a more natural approach in
certain settings. We stress that while restricting the class of invariant measures
using thresholds for the topological pressure or Lyapunov exponents seem to be
natural it is yet to be shown to be a working tool in effecting thermodynamic
formalism.

1.3.4. Outline of the Paper

A direct application of the uniformly hyperbolic approach in the
non-uniformly hyperbolic setting is hopeless in general; we cannot expect to
have finite Markov partitions. Indeed, if a map possesses a Markov
partition, then its topological entropy is the logarithm of an algebraic number,
which should certainly not be expected in general. On the other hand, in the
presence of a hyperbolic invariant measure $ \mu$ of positive entropy,
there are horseshoes
with finite Markov partitions whose entropy approximates the entropy of
$ \mu$ [Katok1980], but these have zero
$ \mu$-measure. ×17 However, in many cases it is
possible to use the symbolic approach by finding a countable Markov partition
, or the related tools of a Young tower
or a more general inducing scheme
; these are discussed in Sects. 2–4. This
approach is challenging to apply completely, but can help establish existence
and uniqueness of equilibrium measures and study their statistical properties
including decay of correlations and the CLT.

A second approach is to avoid the issue of building a Markov
partition by adapting Bowen's specification
property to the non-uniformly hyperbolic setting; this is discussed in Sect.
5. This is similar to the symbolic approach in that
one uses a "coarse-graining" of the system to make counting arguments
borrowed from statistical physics, but sidesteps the issue of producing a
Markov structure. The price paid for this added flexibility is that while
existence and uniqueness can be obtained with specification-based techniques,
there does not seem to be a direct way to obtain strong statistical properties
without first establishing some sort of Markov structure.

A third approach, which we discuss in Sect. 6, is
geometric and is based on pushing forward the leaf volume on unstable
manifolds by the dynamics. More generally, one can work with approximations
to unstable manifolds by admissible
manifolds and use measures which have positive densities with respect to the
leaf volume as reference measures. Such pairs of admissible manifolds and
densities are called standard
and working with them has proven to be quite a useful technique in various
problems in dynamics. This notion was introduced by [Chernov and Dolgopyat2009]. ×18 So far the geometric approach
can be used to establish existence of SRB measures for uniformly hyperbolic
and some non-uniformly hyperbolic attractors and one can also use a version
of this method to construct equilibrium measures for uniformly hyperbolic
sets, see Sect. 6; the questions of uniqueness and statistical
properties using this approach as well as construction of equilibrium measures
for non-uniformly hyperbolic systems are still open.

In the remainder of this paper we describe the three approaches just
listed in more detail, and discuss their application to open problems in the
thermodynamics of non-uniformly hyperbolic systems.

2.1. Earlier Results: One-dimensional and Rational
Maps

In one form or another, the use of Markov models with countably
many states to study non-uniformly hyperbolic systems dates back to the late
1970s and early 1980s, when [Hofbauer1979], [Hofbauer1981a], [Hofbauer1981b] used a countable-state Markov
model to study equilibrium states for piecewise monotonic interval maps.
Indeed, such models for $ \beta$-transformations were studied already in
1973 by [Takahashi1973].

In [Jakobson1981] Jakobson initiated the study of
thermodynamics of unimodal interval maps by constructing absolutely
continuous invariant measures (acim) for the family of quadratic maps
$ f_a(x) = 1-ax^2$ whenever $ a\in\Delta$, where $ \Delta$ is a set of
parameters with positive Lebesgue measure. First we discuss in Sect. 2.2 the
extensions of Jakobson's result to study SRB measures by what have become
known as Young towers
. Then in Sect. 3 we discuss the study of general equilibrium
states in the setting of topological Markov chains with countably many states,
which generalizes the SFT theory from Sect. 1.2. Finally, in
Sect. 4
we discuss the use of inducing schemes
to apply this theory to the thermodynamics of smooth examples.

2.2. Young Towers and
Gibbs–Markov–Young Structures.

2.2.1. Tower Constructions in Dynamical
Systems

Roughly speaking, a tower construction begins with a base
set $ \Lambda$, a map $ G\colon\Lambda\to\Lambda$, and a
height
function $ R\colon\Lambda\to\NN$. Then the tower is constructed as $ \tilde\Lambda:=\{(z,n)\in\Lambda\times \{0,1,2,\dots\}:n< R(z)\}$, and a
map $ g\colon\tilde\Lambda\to\tilde\Lambda$ is defined by $ g(z,n)=(z,n+1)$ whenever $ n+1< R(z)$, and
$ g(z,R(z)-1)=(F(z),0)$. Typically one requires that the dynamics of the return map
$ G$ can be coded by a full shift, or a Markov shift on a countable
set of states. To study a dynamical system $ f\colon X\to X$ using a tower, one
defines a coding map $ \pi\colon\tilde \Lambda\to X$ such that $ f\circ\pi=\pi\circ g$; this coding map
is usually not surjective (the tower does not cover the entire space), and so we
will ultimately need to give some "largeness" condition on the tower. It is
important to distinguish between the case when $ \pi(\Lambda)$ is disjoint from
$ \pi(\tilde\Lambda{\setminus}\Lambda)$, so that the height $ R$ is the first return time to the
base $ \pi(\Lambda)$, and the case when $ R$ is not the first return
time.

Tower constructions for which the height of the tower is the first
return time to the base of the tower are classical objects in ergodic theory and
were considered in works of Kakutani, Rokhlin, and others. Towers for which
the height of the tower is not the first return time appeared in the paper by
[Neveu1969] under the name of temps d'arret
and in the context of dynamical systems in the paper by [Schweiger1975], [Schweiger1979] under the name jump transformation
(which are associated with some fibered systems
; see also the paper by [Aaronson et al.1993] for some general results on
ergodic properties of Markov fibered systems and jump transformations).

A tower construction is implicitly present in Jakobson's proof of
existence of physical measures for quadratic maps. The first significant use of
the tower approach beyond the one-dimensional setting came in the study of
the Hénon map

which for $ b\approx 0$ can be viewed as a two-dimensional extension
of a unimodal map with parameter $ a$. Building on their alternate
proof of Jakobson's theorem in [Benedicks and Carleson1985], Benedicks and
Carleson showed in ([Benedicks and Carleson1991]) that when
$ b$ is sufficiently close to 0, there is a set $ \Delta_b\subset \RR$ of positive
Lebesgue measure such that $ f_{a,b}$ has a topologically transitive
attractor for every $ a\in \Delta_b$. Soon afterwards, Benedicks and Young
established existence of an SRB measure for these examples ([Benedicks and Young1993]); their approach also
gives exponential decay of correlations and the CLT ([Benedicks and Young1995]).

The general structure behind these results was developed in [Young1998] and has come to be known as a Young tower
, It is worth mentioning that a major achievement of [Young1998] was to establish exponential decay of
correlations for billiards with convex scatterers, which is an example of a
uniformly hyperbolic system with discontinuities; we will not discuss such
examples further in this paper. ×19 or a
Gibbs–Markov–Young structure
. The principal feature of a Young tower is that the induced map on the base
of the tower is conjugate to the full shift on the space of two-sided sequences
over countable alphabet. This allows one to use some recent results on
thermodynamics of this symbolic map to establish existence and uniqueness of
equilibrium measures for the original map and study their ergodic properties.

2.2.2. Young Diffeomorphisms

A $ C^{1+\alpha}$ diffeomorphism $ f$ of a compact smooth
manifold $ M$ is called Young
diffeomorphism
if it admits a Young tower
. This tower has a particular structure which is characterized as follows:

The base $ \Lambda$ of the tower has hyperbolic product structure
which is generated by continuous families $ {\bf V}^u=\{V^u\}$ and $ {\bf V}^s=\{V^s\}$ of
local unstable and stable manifolds.

The induced map has
the Markov property, is uniformly hyperbolic and has uniform bounded
distortion.

The intersection of at least one unstable
manifold with the base of the tower has positive leaf volume It follows
that every local unstable manifold intersects the base in a set of positive leaf
volume. ×20 and the integral of the height of
the tower against leaf volume is finite.

In particular, the tower codes a positive volume part of the system (but not
necessarily all trajectories) by a countable state Markov shift.

A formal description of the Young tower is as follows. There are two
continuous families $ {\bf V}^u=\{V^u\}$ and $ {\bf V}^s=\{V^s\}$ of local unstable and
stable manifolds, respectively, with the property that each $ V^s$
meets each $ V^u$ transversely in a single point and $ \Lambda=(\bigcup V^u)\cap (\bigcup V^s)$; a
union of some of the manifolds $ V^u$ is called a $ u$ -set
, a union of some of the manifolds $ V^s$ is called an $ s$
-set
. One asks for $ \Lambda$ to have the following properties; here
$ C,\eta> 0$ and $ \beta\in (0,1)$ are constants.

( P3
) Defining the recurrence (induced) time
$ R\colon\bigcup_i\Lambda_i^s\to\Lambda$ by $ R|\Lambda_i^s=R_i$ and the induced
map
$ F(x) = f^{R(x)}(x)$, we have that for all $ n\ge 1$

Forward contraction on
$ V^s$: if $ x,y$ are in the same leaf $ V^s$, then
$ d(F^nx,F^ny)\le C\beta^nd(x,y)$.

Backward
contraction on
$ V^u$: if $ x,y$ are in the same leaf $ V^u$ and the
same $ s$-set $ \Lambda_i^s$, then $ d(F^{-n}x,F^{-n}y)\le C\beta^nd(Fx,Fy)$.

Bounded distortion:
if $ x,y$ are in the same leaf $ V^u$ and the same
$ s$-set $ \Lambda_i^s$ then \begin{eqnarray*} \log\frac{|\det dF^u(x)|}{|\det dF^u(y)|}\le C d(Fx,Fy)^\eta. \end{eqnarray*}

Our description of Young tower follows [Pesin et al.2016b] and differs from the original
description in [Young1998]. Most importantly, we do not require
that the map $ f$ contracts distances along local stable manifolds
uniformly with an exponential rate and neither does the inverse map
$ f^{-1}$ along local unstable manifolds but that this requirement holds
with respect to the induced map $ F$ (see ( P3
)). We stress that in constructing SRB and equilibrium measures on Young
towers and studying their ergodic properties these extra requirements on the
maps $ f$ and $ f^{-1}$ are not needed and that there are
examples in which the map $ f$ contracts distances along local
stable manifolds uniformly with a polynomial rate, see Sect. 2.3.2.

2.2.3. SRB Measures for
Young Diffeomorphisms.

Once a tower structure has been found, the strength of the
conclusions one can draw depends on the rate of decay of the tail of the tower
; that is, the speed with which $ m_{V^u}\{x\in V^u\mid R(x) > T\}\to 0$ as $ T\to\infty$ for
$ V^u\in {\bf V}^u$. We say that with respect to the measure $ m_{V^u}$ the
tower has

To describe ergodic properties of the SRB measure one needs an extra
condition. We say that the tower satisfies the
arithmetic condition
if the greatest common denominator of of the set of integers $ \{R_i\}$ is
one. The tower $ \tilde\Lambda$ admits a natural countable
Markov partition (see [Young1998]) and the arithmetic condition is
equivalent to the requirement that the corresponding Markov shift is
topologically mixing. ×21

Theorem 2.2.

([Young1998]) Let $ f$ be a $ C^{1+\alpha}$
diffeomorphism of a compact manifold $ M$ admitting a Young
tower. Assume that the tower satisfies
(2.3)
, the arithmetic condition and has exponential (respectively, polynomial) tails.
Then $ (f,\mu)$ is Bernoulli, has exponential (respectively, polynomial)
decay of correlations and satisfies the CLT with respect to the class of
functions which are Hölder continuous on $ \Lambda$.

Note that even without the arithmetic condition one still obtains the
"exponential decay up to a period" result stated earlier in Theorem 1.1
(2)
.

In [Young1999], Young gave an extension of the results
from [Young1998] that applies in a more abstract setting,
giving existence of an invariant measure that is absolutely continuous with
respect to some reference measure (not necessarily Lebesgue). She also
provided a condition on the height of the tower that guarantees a polynomial
upper bound for the decay of correlations. The corresponding polynomial
lower bound (showing that Young's bound is optimal) was obtained by [Sarig2002] and [Gouëzel2004].

The flexibility in the reference measure makes Young's result suitable
for studying existence, uniqueness and ergodic properties of equilibrium
measures other than SRB measures (although this was not done in [Young1999]). In particular, this is used in the
proof of Statement 2 of Theorem 2.3 below; we discuss such questions more in Sects.
3, 4.

Just as the Hénon maps can be studied as a "small"
two-dimensional extension of the unimodal maps, Theorems 2.1 and 2.2 can be
applied to more general 'strongly dissipative' maps that are obtained as 'small'
two-dimensional extensions of one-dimensional maps; this is carried out in
[Wang and Young2001], [Wang and Young2008].

Aside from such strongly dissipative maps, Young towers have been
constructed for some partially hyperbolic maps where the center direction is
non-uniformly contracting ([Castro2004]) or expanding ([Alves and Pinheiro2010], [Alves and Li2015]); the latter papers are built on
earlier results for non-uniformly expanding maps where one does not need to
worry about the stable direction ([Alves et al.2005], [Gouëzel2006]). In both cases existence (and
uniqueness) of an SRB measure was proved first ([Bonatti and Viana2000], [Alves et al.2000]) via other methods closer to the
push-forward geometric approach that we discuss in Sect. 6, so the
achievement of the tower construction was to establish exponential decay of
correlations and the CLT. These results only cover the SRB measure and do
not consider more general equilibrium states.

2.2.4. Thermodynamics of Young
Diffeomorphisms for the Geometric $ t$-Potential

Let $ f$ be a $ C^{1+\alpha}$ Young diffeomorphism of a
compact smooth manifold $ M$. Consider the set $ \Lambda$ with
hyperbolic product structure. Let $ \Lambda_i^s$ be the collections of
$ s$-sets and $ R_i$ the corresponding inducing times. Set
\begin{eqnarray*} Y=\bigcup_{k\ge 0}\,f^k(\Lambda). \end{eqnarray*} This is a forward invariant set for $ f$. For every
$ y\in Y$ the tangent space at $ y$ admits an invariant splitting
$ T_yM=E^s(y)\oplus E^u(y)$ into stable and unstable subspaces. Thus we can consider the
geometric $ t$-potential $ \varphi_t(y)$ which is well defined for
$ y\in Y$ and is a Borel (but not necessarily continuous) function for
every $ t\in\RR$. We consider the class $ \mathcal{M}(f,Y)$ of all invariant
measures $ \mu$ supported on $ Y$, i.e., for which
$ \mu(Y)=1$. It follows that $ \mu(\Lambda)> 0$, so that $ \mu$ 'charges'
the base of the Young tower. Further, given a number $ h> 0$, we
denote by $ \mathcal{M}(f,Y, h)$ the class of invariant measures $ \mu\in\mathcal{M}(f,Y)$ for
which $ h_\mu(f)> h$.

The following result describes existence, uniqueness, and ergodic
properties of equilibrium measures. Given $ n> 0$, denote by
\begin{eqnarray*} S_n:=\text{Card}\{\Lambda_i^s\colon R_i=n\}. \end{eqnarray*}

Theorem 2.3.

(see [Pesin et al.2016b, Melbourne and Terhesiu2014])
Assume that the Young tower satisfies:

(1) for all large
$ n$

\begin{eqnarray}\label{growth} S_n\le e^{hn}, \end{eqnarray}

(2.4)

where $ 0< h< h_{\mu_1}(f)$ is a constant and $ \mu_1$ is the SRB
measure for $ f$;

(2) the set $ \bigcup_{i\ge 1}(\bar{\Lambda}_i{\setminus}\Lambda_i)$ supports no invariant
measure that gives positive weight to any open set. This condition is stronger
than the corresponding condition
(2.3)
. ×22

Then there is $ t_0< 0$ such that for $ t_0\le t< 1$ there exists a measure
$ \mu_t$ which is a unique equilibrium measure for $ \varphi_t$ among
all liftable
measures (see the remark below). If in addition, the tower satisfies the
arithmetic condition, This requirement should be added to Theorem 4.5,
Statement 2 of Theorem 4.7 and Statement 3 of Theorem 7.1 in [Pesin et al.2016b]. ×23 then $ (f,\mu_t)$ is Bernoulli,
has exponential decay of correlations and satisfies the CLT with respect to a
class of potential functions which contains all Hölder continuous
functions on $ Y$.

Remark.

1. The requirement
(2.4)
means that the number of $ s$-sets in the base of the tower can grow
exponentially but with rate slower than the metric entropy of the SRB
measure. This is a strong requirement on the Young tower, but it is known to
hold in some examples, see Sect. 2.3 below.

2. For
$ t=1$, the SRB measure $ \mu_1$ may not have exponential
decay of correlations; this is the case for the Manneville–Pomeau map
where the decay is polynomial. See Sect. 1.3.2 and
also Sect. 2.3 for more details.

3. We stress
that the measures $ \mu_t$ are equilibrium measures within the class of
measures that can be lifted
to the tower: recall that an invariant measure $ \mu$ supported on
$ Y$ is called liftable
if there is a measure $ \nu$ supported on $ \Lambda$ and invariant
under the induced map $ F$ such that the number

In particular, $ \mu_t={\mathcal{L}}(\nu_t)$ for some measure $ \nu_t$ which is
an equilibrium (and indeed, Gibbs) measure for the induced map
$ F$.

Under the condition 2.4 every measure with entropy $ {> }h$ is
liftable. In general, it is shown in [Zweimüller2005] that if $ R\in L^1(Y,\mu)$ then
$ \mu$ is liftable. In particular, if the return time $ R$ is the
first return time to the base of the tower, then every measure that charges the
base of the tower is liftable.

4. The proof of
exponential decay of correlations and the CLT is based on showing the
exponential tails property of the measure $ \nu_t$ See
(2.2)
where one should replace the leaf volume with the measure $ \nu_t$.
×24 (see [Pesin et al.2016b], Theorem 4.5) and then applying
results from [Melbourne and Terhesiu2014]. In [Melbourne and Terhesiu2014] the authors
considered only expanding maps and Young towers with polynomial tails,
however, their results can easily be extended to invertible maps and Young
towers with exponential tails. ×25

5. For a
$ C^{1+\alpha}$ diffeomorphism $ f$ there may exist several Young
towers with bases $ \Lambda_k$, $ k=1,\dots,m$, such that the corresponding
sets $ Y_k$ are disjoint. For each $ k$, Theorem 2.3 gives a number
$ t_{0k}< 0$ and for every $ t_{0k}< t< 1$ the equilibrium measure
$ \mu_{tk}$ for the geometric potential $ \varphi_t$. This measure is
unique within the class of measures $ \mu$ for which $ \mu(Y_k)=1$ and
$ h_\mu(f)> h$ where $ 0< h< h_{\mu_1}(f)$. Note that both $ h$ and
$ h_{\mu_1}(f)$ do not depend on $ k$. ×26 Setting $ t_0=\max_{1\le k\le m}t_{0k}$, for every
$ t_0< t< 1$ we obtain the measure $ \mu_t$ such that $ \mu_t|Y_k=\mu_{tk}$.
If for every measure $ \mu$ with $ h_\mu(f)> h$, we have that
$ \mu(Y_k)> 0$ for some $ 1\le k\le m$, then the measure $ \mu_t$ is the
unique equilibrium measure for $ \varphi_t$ within the class of invariant
measures with large entropy. This is the case in the two examples described in
Sect. 2.3.

6. It is known
that $ t=1$ can be a phase transition, that is the pressure function
$ P(t)$ is not differentiable and there are more than one equilibrium
measures for $ \varphi_1$. However, it is not known whether phase
transitions can occur for $ t< t_0$.

7. Theorem
2.3 is a
corollary of a more general result establishing thermodynamics for maps
admitting inducing schemes of hyperbolic type, see Theorem 4.1.

2.3. Examples of Young
Diffeomorphisms.

We describe two examples of Young diffeomorphisms for which
Theorem 2.3
applies.

2.3.1. A Hénon-like Diffeomorphism at
the First Bifurcation

The first example is Hénon-like diffeomorphisms of the plane
at the first bifurcation parameter. For parameters $ a,b$ consider the
Hénon map $ f_{a,b}$ given by
(2.1)
. It is shown in [Bedford and Smillie2004], [Bedford et al.2006], [Cao et al.2008] that for each $ 0< b\ll 1$ there
exists a uniquely defined parameter $ a^*=a^*(b)$ such that the
non-wandering set for $ f_{a,b}$ is a uniformly hyperbolic horseshoe for
$ a > a^*$ and the parameter $ a^*$ is the first parameter value for
which a homoclinic tangency between certain stable and unstable manifolds
appears.

Theorem 2.4.

([Senti and Takahasi2013], [Senti and Takahasi2016], Theorem A) For any
bounded open interval $ I\subset(-1,+\infty)$ there exists $ 0< b_0\ll 1$ such that if
$ 0\le b< b_0$ then

(1) the map
$ f_{a^*(b), b}$ is a Young diffeomorphism;

(2) there exists a unique equilibrium measure for the
geometric $ t$-potential and for all $ t\in I$.

2.3.2. The Katok
Map.

We describe the Katok map ([Katok1979]) (see also [Barreira and Pesin2013]), which can be thought of
as an invertible and two-dimensional analogue of the
Manneville–Pomeau map. Consider the automorphism of the 2-torus
given by the matrix $ T=( \begin{array}{ll} 2 &\quad 1\\ 1 &\quad 1\end{array})$ and then choose $ 0< \alpha< 1$ and a
function $ \psi:[0,1]\mapsto[0,1]$ satisfying:

$ \psi$ is of class $ C^\infty$ except at zero;

$ \psi(u)=1$ for $ u\ge r_0$ and some $ 0< r_0< 1$;

$ \psi'(u)> 0$ for every $ 0< u< r_0$;

$ \psi(u)=(ur_0)^\alpha$ for $ 0\le u\le\frac{r_0}{2}$.

Let $ D_r=\{(s_1,s_2): {s_1}^2+{s_2}^2\le r^2\}$ where $ (s_1,s_2)$ is the coordinate system obtained from
the eigendirections of $ T$. Consider the system of differential
equations in $ D_{r_0}$

where $ \lambda> 1$ is the eigenvalue of $ T$. Observe that
$ T$ is the time-1 map of the flow generated by the system of
equations
(2.7)
.

We slow down trajectories of
(2.7)
by perturbing it in $ D_{r_0}$ as follows: \begin{eqnarray*} \dot{s}_1=s_1\psi({s_1}^2+{s_2}^2)\log\lambda, \quad \dot{s}_2=- s_2\psi({s_1}^2+{s_2}^2)\log\lambda. \end{eqnarray*} This generates a
local flow, whose time-1 map we denote by $ g$. The choices of
$ \psi$ and $ r_0$ guarantee that the domain of $ g$
contains $ D_{r_0}$. Furthermore, $ g$ is of class $ C^\infty$
in $ D_{r_0}$ except at the origin and it coincides with $ T$ in
some neighborhood of the boundary $ \partial D_{r_0}$. Therefore, the map
\begin{eqnarray*} G(x)=\begin{cases} T(x) & \text{if} \quad x\in\mathbb{T}^2{\setminus} D_{r_0},\\ g(x) & \text{if} \quad x\in D_{r_0} \end{cases} \end{eqnarray*} defines a homeomorphism of the torus, which is a $ C^\infty$
diffeomorphism everywhere except at the origin.

The map $ G$ preserves the probability measure
$ d\nu=\kappa_0^{-1}\kappa\,dm$ where $ m$ is the area and the density $ \kappa$
is defined by \begin{eqnarray*} \kappa(s_1,s_2):=\begin{cases} (\psi({s_1}^2+{s_2}^2))^{-1} &\text{if}\,\, (s_1,s_2)\in D_{r_0},\\ 1 & \text{otherwise} \end{cases} \end{eqnarray*} and \begin{eqnarray*} \kappa_0:=\int_{\mathbb{T}^2}\kappa\,dm. \end{eqnarray*} We further perturb the map
$ G$ by a coordinate change $ \phi$ in $ \mathbb{T}^2$ to
obtain an area-preserving $ C^\infty$ diffeomorphism. To achieve this,
define a map $ \phi$ in $ D_{r_0}$ by the formula

and set $ \phi=\text{Id}$ in $ \mathbb{T}^2{\setminus} D_{r_0}$. Clearly, $ \phi$ is a
homeomorphism and is a $ C^\infty$ diffeomorphism outside the origin.
One can show that $ \phi$ transfers the measure $ \nu$ into the
area and that the map $ f=\phi\,\circ\, G\,\circ\,\phi^{-1}$ is a $ C^\infty$ diffeomorphism. This
is the Katok map ([Katok1979], [Barreira and Pesin2013]). One can show that the
map $ f$ has nonzero Lyapunov exponents almost everywhere.
However, there are trajectories with zero Lyapunov exponents, for example
the origin is a neutral fixed point. ×27

Theorem 2.5.

(see [Pesin et al.2016a]) The following statements hold:

(1) the Katok map
$ f$ is a Young diffeomorphism; moreover,

there are finitely many disjoint sets $ \Lambda_k$ that are
bases of Young towers for which the corresponding sets $ Y_k$ cover
the whole torus except for the origin;

every invariant
measure $ \mu$ except for the Dirac measure at the origin
$ \delta_0$ can be lifted to one of the towers.

(2) For any
$ t_0< 0$ one can find a small $ r_0=r_0(t_0)$ such that if the construction
is carried out with this value of $ r_0$, then for every $ t_0< t< 1$

The thermodynamic formalism for SFTs rested on the Ruelle's
version of the Perron–Frobenius theorem for finite-state topological
Markov chains. For the class of two-step potential functions $ \ph(x) = \ph(x_0,x_1)$,
which includes the zero potential $ \ph=0$, the extension of this theory to
countable-state Markov shifts dates back to work of [Vere-Jones1962], [Vere-Jones1967], [Gurevič1969], [Gurevič1970], [Gurevič1984], and [Gurevich and Savchenko1998]; we discuss this in
Sect. 3.1.
For more general potential functions a sufficiently complete picture is
primarily due to Sarig, and we discuss these in Sect. 3.2.

3.1. Recurrence Properties for
Random Walks.

Recall the form of Theorem 1.3 on existence of a unique MME for SFTs:

(1) the largest
eigenvalue $ \lambda$ of the transition matrix $ A$ determines
the topological entropy;

(3) $ P$ has a unique stationary vector
$ \pi$ (which can be written in terms of left and right eigenvectors for
$ (A,\lambda)$), which determines a Markov measure that is the unique MME.

In the countable-state setting, existence of eigenvectors and stationary vectors
is a more subtle question (although once these are found, the proof of
uniqueness goes through just as in the finite-state case). The general story is
well-illustrated by just considering the last step above: suppose we are given a
stochastic matrix $ P_{ij}$ with countably many entries. This
corresponds to a directed graph $ G$ with countably many vertices,
whose edges are given weights as follows: the weight of the edge from
$ i$ to $ j$ is $ P_{ij}$. Then one can consider the
Markov chain described by $ P$ as a
random walk
on $ G$.

Existence of a stationary vector $ \pi=(\pi_i)$ with $ \pi P=\pi$ is
determined by the recurrence
properties of the shift [Vere-Jones1962], [Vere-Jones1967]. Suppose we start our random
walk at a vertex $ a$; one can show that the probability that we
return to $ a$ infinitely many times is either 0 or 1. If the
probability of returning infinitely many times is 1, then the walk is recurrent
. Recurrence is necessary in order to have a stationary probability vector
$ \pi$, but it is not sufficient; one must distinguish between the case
when our expected return time is finite ( positive
recurrence
) and when it is infinite ( null recurrence
). If the walk is positive recurrent then there is a stationary probability vector
$ \pi$; if it is null recurrent then one can still find a vector
$ \pi$ such that $ \pi P=\pi$, but one has $ \sum_i\pi_i=\infty$, so
$ \pi$ cannot be normalized to a probability vector.

In fact, the trichotomy between transience, null recurrence, and
positive recurrence is the key to generalizing all of Theorem 1.3 to the
countable-state case [Pesin2014]. The recurrence conditions can be
formulated in terms of the number of loops in the graph $ G$. Fixing
a vertex $ a$, let $ Z_n^*$ be the number of simple
loops of length $ n$ based at $ a$ (first returns to
$ a$) and $ Z_n$ be the number of
all
loops of length $ n$ based at $ a$ (including loops which
return more than once). In the next section when we consider non-zero
potentials, we will have to count the loops with weights coming from the
potential. ×28

(1) The supremum
of the metric entropies is equal to the Gurevich
entropy
$ h_G :=\lim\frac1n\log Z_n$ (the limit exists if the graph is aperiodic; otherwise one should
take the upper limit).

(2) The shift $ \Sigma_A$ is recurrent
if $ \sum_n e^{-nh_G} Z_n = \infty$, and transient
if the sum is finite. For some intuition behind this definition, it may be
helpful to consider again a countable-state random walk: writing
$ \PP(n)$ for the probability of returning to the original vertex at time
$ n$, we recall that by the Borel–Cantelli lemma, the walk is
recurrent (infinitely many returns a.s.) if $ \sum_n \PP(n)=\infty$, and transient (finitely
many returns a.s.) if the sum is finite. ×29 The eigenvectors $ h$
and $ v$ for $ (A,\lambda)$ exist if and only if $ \Sigma_A$ is
recurrent.

(3)
Among recurrent shifts, one must distinguish between positive recurrence
, when $ \sum_n n e^{-nh_G} Z_n^* < \infty$, and null recurrence
, when the sum diverges. Writing $ \pi_i = h_i v_i$, one has $ \sum \pi_i < \infty$ if
$ \Sigma_A$ is positive recurrent (hence, $ \pi$ can be normalized),
and $ \sum \pi_i = \infty$ if it is null recurrent. One can also characterize positive
recurrent shifts as those for which $ e^{nh_G} Z_n$ is bounded away from 0 and
$ \infty$, which immediately implies divergence of the sum
$ \sum_n e^{-nh_G} Z_n$, while null recurrent shifts are those for which $ \llim_n e^{-nh_G} Z_n = 0$
but the sum still diverges.

It is instructive to note that once a distinguished vertex
$ a$ is fixed as the starting point of the loops, one can view the first
return map to $ [a]$ as a Young tower, and then the summability
condition in positive recurrence is equivalent to the condition that the tails of
the tower are integrable, which was the existence criterion in Theorem 2.1.

3.2. Non-zero
Potentials.

In discussing the extension to non-zero potentials on countable-state
topological Markov chains, we will follow the notation, terminology, and
results of [Sarig1999], [Sarig2001b], [Sarig2001a], although the contributions of [Gurevič1984], [Gurevich and Savchenko1998], [Mauldin and Urbański1996], [Mauldin and Urbański2001], [Aaronson and Denker2001], and of [Fiebig et al.2002] should also be mentioned. Sarig
adapted transience, null recurrent, and positive recurrence for non-zero
potential functions. The summability criterion for positive recurrence is
exactly as above, except that now $ Z_n$ represents the total weight of
all loops of length $ n$ and $ Z_n^*$ represents the total weight
of simple loops of length $ n$ where weight is computed with respect
to the potential function; more precisely \begin{eqnarray*} Z_n=Z_n(\varphi,a)=\sum_{\sigma^n(x)=x}\exp(\Phi_n(x))1_{[a]}(x) \end{eqnarray*} and \begin{eqnarray*} Z_n^*=Z_n^*(\varphi,a)=\sum_{\sigma^n(x)=x}\exp(\Phi_n(x))1_{[\varphi_a=n]}(x), \end{eqnarray*} where
$ \Phi_n(x)=\sum_{k=0}^{n-1}\varphi(f^kx)$. Furthermore, the Gurevich entropy $ h_G(\sigma)$ is replaced
with the Gurevich-Sarig pressure
$ P_{GS}(\sigma,\ph)$, which is the exponential growth rate of $ Z_n$, i.e.,
\begin{eqnarray*} P_{GS}(\sigma,\ph)=\lim_{n\to\infty}\frac1n \log Z_n. \end{eqnarray*} For Markov shifts with finite topological entropy, [Buzzi and Sarig2003] proved that an equilibrium
measure exists if and only if the shift is positive recurrent. A good summary of
the theory can be found in [Sarig2015]. For our purposes the main result is the
following.

Theorem 3.1.

Let $ \Sigma$ be a topologically mixing
countable-state Markov shift with finite topological entropy, and let
$ \ph\colon X\to\RR$ be a Hölder continuous In fact Theorem 3.1 holds
for the more general class of potentials with
summable variations
, but Hölder continuity is needed for the statistical properties mentioned
below. ×30 function such that $ P_{GS}(\ph)< \infty$.
Then $ \ph$ is positive recurrent if and only if there are $ \lambda> 0$,
a positive continuous function $ h$, and a conservative measure
$ \nu$ (i.e., a measure that allows no nontrivial wandering sets) which
is finite and positive on cylinders, such that $ \LLL_\ph h=\lambda h$, $ \LLL_\ph^*\nu=\lambda\nu$, and
$ \int h\,d\nu=1$. In this case the following are true.

The statistical properties of $ \mu$ depend
on the rate of convergence in the last item of Theorem 3.1, which
in turn depends on how quickly $ Z_n^* e^{-nP_{GS}}$ goes to 0. If it goes to zero with
polynomial rate then the corresponding tower (obtained by inducing on a
single state) has polynomial tails, and the equilibrium state has polynomial
decay of correlations. If it goes to zero with exponential speed—that is,
if $ Z_n^*$ has smaller exponential growth rate than
$ Z_n$—then the tower has exponential tails and correlations
decay exponentially. In this case the shift is called strong positive recurrent
; see [Cyr and Sarig2009] for a summary of the results in
this case.

3.3. Countable-State Markov Partitions for Smooth
Systems

Using Pesin theory, Sarig recently carried out a version of the
construction of Markov partitions for non-uniformly hyperbolic
diffeomorphisms in two dimensions. Recall that for a uniformly hyperbolic
diffeomorphism $ f\colon M\to M$, one obtains an SFT $ \Sigma$ and a
coding map $ \pi\colon \Sigma\to M$ such that

$ \pi$ is Hölder continuous and has
$ f\circ \pi = \pi \circ \sigma$;

$ \pi$ is onto and is 1–1 on
a residual set $ \Sigma' \subset \Sigma$ that has full measure for every equilibrium state
of a Hölder potential on $ \Sigma$.

In non-uniform hyperbolicity one must replace the SFT with a countable-state
Markov shift, and also weaken some of the conclusions.

if $ \mu$ is an ergodic
$ f$-invariant measure on $ M$ with $ h_\mu(f) > \chi$, then
$ \mu(\pi(\Sigma_\chi))=1$, and moreover there is an ergodic $ \sigma$-invariant
measure $ {\hat{\mu}}$ on $ \Sigma_\chi$ such that $ (\pi_\chi)_* {\hat{\mu}} = \mu$ and
$ h_{{\hat{\mu}}}(\sigma) = h_\mu(f)$.

Observe that Theorem 3.2 echoes our
recurring theme that in non-uniform hyperbolicity, to obtain 'good'
hyperbolic-type results one often needs to ignore a 'small-entropy' part of the
system. In fact the key property of the threshold $ \chi$ is that by the
Margulis–Ruelle inequality, any ergodic measure with $ h_\mu(f)> \chi$
must have positive Lyapunov exponent at least $ \chi$. Thus for a
higher-dimensional generalization of Theorem 3.2, one should
expect that the natural condition would be on the Lyapunov exponents, rather
than the entropy.

The analogous result to Theorem 3.2 for
three-dimensional flows was proved by [Lima and Sarig2014]. In both cases this can be
used to deduce Bernoullicity up to finite rotations of ergodic positive entropy
equilibrium states ([Sarig2011], [Ledrappier et al.2016]). However, these general
results do not give any information on the recurrence properties of the
countable state shift, or the tail of the resulting tower, and in particular they
do not provide a mechanism for verifying decay of correlations and the CLT.
This is of no surprise, since at this level of generality, one should not expect to
get exponential decay (or any other particular rate).

The study of SRB measures via Young towers generalizes to the
study of equilibrium states via inducing schemes
, which use the tower approach to model (a large part of) the system by a
countable-state Markov shift, and then apply the thermodynamic results from
Sect. 3.
The concept of an inducing scheme in dynamics is quite broad and applies to
systems which may be invertible or not, smooth or not differentiable. Every
inducing scheme generates a symbolic representation by a tower which is well
adapted to constructing equilibrium measures for an appropriate class of
potential functions using the formalism of countable state Markov shifts. The
projection of these measures from the tower are natural candidates for the
equilibrium measures for the original system.

In order to use this symbolic approach to establish existence and to
study equilibrium states, some care must be taken to deal with the liftability problem
as only measures that can be lifted to the tower can be 'seen' by the tower.

One may consider inducing schemes of expanding type, or of
hyperbolic type. The former were introduced in [Pesin and Senti2008] and apply to study
thermodynamics of non-invertible maps (e.g., non-uniformly expanding maps)
while the latter were introduced in [Pesin et al.2016b] and are used to model invertible
maps (e.g,, non-uniformly hyperbolic maps). In this paper we only consider
inducing schemes of hyperbolic types and we follow [Pesin et al.2016b].

Let $ f\colon X\to X$ be a homeomorphism of a compact metric space
$ (X, d)$. We assume that $ f$ has finite topological entropy
$ \htop(f)< \infty$. An inducing scheme of hyperbolic
type
for $ f$ consists of a countable collection of disjoint Borel sets
$ S=\{J\}$ and a positive integer-valued function $ \tau\colon S\to\mathbb{N}$; the
inducing domain
of the inducing scheme $ \{S,\tau\}$ is $ W=\bigcup_{J\in S}J$, and the inducing time
$ \tau\colon X\to\mathbb{N}$ is defined by $ \tau(x)=\tau(J)$ for $ x\in J$ and $ \tau(x)=0$
otherwise. We require several conditions.

( I1
) For any $ J\in S$ we have $ f^{\tau(J)}(J)\subset W$ and $ \bigcup_{J\in S}f^{\tau(J)}(J)=W$.
Moreover, $ f^{\tau(J)}|J$ can be extended to a homeomorphism of a
neighborhood of $ J$.

This condition allows one to define the induced
map
$ F\colon W\to W$ by setting $ F|J:=f^{\tau(J)}|J$ for each $ J\in S$. If
$ \tau$ is the first return time to $ W$, then all images
$ f^{\tau(J)}(J)$ are disjoint. However, in general the sets $ f^{\tau(J)}(J)$
corresponding to different $ J\in S$ may overlap. In this case the map
$ F$ may not be invertible.

( I2
) For every bi-infinite sequence $ \underline{a}=(a_n)_{n\in\mathbb{Z}}\in S^{\mathbb{Z}}$ there exists a unique
sequence $ \underline{x}=\underline{x}(\underline{a})=(x_n=x_n(\underline{a}))_{n\in\mathbb{Z}}$ such that

(b) if
$ x_n(\underline{a})=x_n(\underline{b})$ for all $ n\le 0$ then $ \underline{a}=\underline{b}$.

This condition allows one to define the coding
map
$ \pi\colon S^{\mathbb{Z}}\to \bigcup\overline{J}$ by $ \pi(\underline{a}):=x_0(\underline{a})$. Within the full shift $ \sigma\colon S^\ZZ\to S^\ZZ$ we
consider the set \begin{eqnarray*} \check{S}:=\{\underline{a}\in S^\ZZ \mid x_n(\underline{a})\in J_{a_n}\text{ for all } n\in\mathbb{Z}\}. \end{eqnarray*} For any $ \underline{a}\in S^\mathbb{Z}{\setminus}\check{S}$ there exists $ n\in\mathbb{Z}$
such that $ \pi\circ\sigma^n(\underline{a})\in\overline{J_{a_n}}{\setminus} J_{a_n}$. In particular, if all $ J\in S$ are closed then
$ S^\mathbb{Z}{\setminus}\check{S}=\emptyset$; however, this need not always be the case. It follows from (
I1
) and ( I2
) that the map $ \pi$ has the following properties:

(1) $ \pi$
is well defined, continuous and for all $ \underline{a}\in S^\mathbb{Z}$ one has $ \pi\circ\sigma(\underline{a})=f^{\tau(J)}\circ\pi(\underline{a})$
where $ J\in S$ is such that $ \pi(\underline{a})\in \bar{J}$;

(2) $ \pi$ is one-to-one on
$ \check S$ and $ \pi(\check S)=W$;

(3) if $ \pi(\underline{a})=\pi(\underline{b})$ for some $ \underline{a}, \underline{b}\in\check{S}$ then
$ a_n=b_n$ for all $ n\ge 0$.

Proving the existence and uniqueness of equilibrium measures requires some
additional condition on the inducing scheme $ \{S,\tau\}$:

( I3
) The set $ S^\mathbb{Z}{\setminus}\check S$ supports no (ergodic)
$ \sigma$-invariant measure which gives positive weight to any open
subset.

This condition is designed to ensure that every equilibrium measure for the
shift is supported on $ \check{S}$ and its projection by $ \pi$ is thus
supported on $ W$ and is $ F$-invariant. This projection is
a natural candidate for the equilibrium measure for $ F$.

Set $ Y= \{f^k(x)\mid x\in W, 0\le k\le \tau(x)-1\}$. Note that $ Y$ is forward invariant
under $ f$. This can be thought of as the region of $ X$
that is 'swept out' as $ W$ is carried forward under the dynamics of
$ f$; in particular, it contains all trajectories that intersect the base
$ W$.

Let $ \varphi$ be a potential function. Existence of an
equilibrium measure for $ \varphi$ is obtained by first studying the
problem for the induced system $ (F,W)$ and the induced potential
$ \overline{\varphi}\colon W\to\mathbb{R}$ defined by
(1.2)
. The study of existence and uniqueness of equilibrium measures for the
induced system $ (F,W)$ is carried out by conjugating the induced
system to the two-sided full shift over the countable alphabet $ S$.
This requires that the potential function $ \Phi:={\bar{\varphi}}\circ \pi$ be well defined on
$ S^\mathbb{Z}$. To this end we require that

( P1
) the induced potential $ \overline{\varphi}$ can be extended by continuity to a
function on $ \bar{J}$ for every $ J\in S$.

Theorem 4.2.

the tower has exponential
tails with respect to the measure $ \nu_{\varphi^+}$ that is there exist
$ C> 0$ and $ 0< \theta< 1$ such that for all $ n> 0$, \begin{eqnarray*} \nu_{\varphi^+}(\{x\in W: \tau(x)\ge n\})\le C\theta^n; \end{eqnarray*}
(compare to
(2.2)
);

the tower satisfies the arithmetic condition. This
requirement should be added to Theorem 4.6 in [Pesin et al.2016b]. ×31

Then $ (f,\mu_\varphi)$ has exponential decay of correlations and satisfies the
CLT with respect to the class of functions whose induced functions on
$ W$ are bounded locally Hölder continuous.

We describe some verifiable
conditions on the potential function $ \varphi$ under which the
assumptions of Theorem 4.1 hold:

(1) there exists a
unique equilibrium measure $ \mu_\varphi$ for $ \varphi$ among all
measures in $ \mathcal{M}_L(f, Y)$; the measure $ \mu_\varphi$ is ergodic;

(2) if $ \nu_{\varphi^+}=\mathcal{L}^{-1}(\mu_\varphi)$
has exponential tail and the tower satisfies the arithmetic condition, then
$ (f,\mu_\varphi)$ has exponential decay of correlations and satisfies the CLT
with respect to a class of functions whose corresponding induced functions on
$ W$ (see
(1.2)
) are bounded locally Hölder continuous functions.

5.
Coarse-Graining, Expansivity, and Specification

5.1. Uniform Expansivity and
Specification

Let $ X$ be a compact metric space and $ f\colon X\to X$ a
homeomorphism; given $ \eps> 0$ and $ x\in X$, the set

contains all points whose trajectory stays within $ \eps$ of
the trajectory of $ x$ for all time. The map $ f$ is
expansive
if there is $ \eps> 0$ such that $ \Gamma_\eps(x) = \{x\}$ for every $ x\in X$;
that is, if any two distinct trajectories eventually separate at scale
$ \eps$. Uniformly hyperbolic systems can easily be shown to be
expansive, and expansivity is a sufficient condition for existence
of an equilibrium measure for any continuous potential $ \ph$; indeed,
the standard proof of the variational principle ([Walters1982], Theorem 8.6) gives a construction of
such a measure. The idea is that one "coarse-grains" the system at scale
$ \eps$ and builds a measure that is appropriately distributed over all
trajectories that separate by $ \eps$ within $ n$ iterates;
sending $ n\to\infty$ and using expansivity one guarantees that this
measure is an equilibrium state.

To show that this equilibrium state is
unique
, Bowen used the following specification property
of uniformly hyperbolic systems: for every $ \eps> 0$ there is
$ \tau\in \NN$ such that any collection of finite-length orbit segments can be
$ \eps$-shadowed by a single orbit that takes $ \tau$ iterates
to transition from one segment to the next. More precisely, if we associate
$ (x,n) \in X\times N$ to the orbit segment $ x,f(x),\dots, f^{n-1}(x)$ and write \begin{eqnarray*} B_n(x,\eps) = \{y\in X \mid d(f^kx,f^ky) \leq \eps \text{ for all } 0\leq k < n\} \end{eqnarray*} for
the Bowen ball
of points that shadow $ (x,n)$ to within $ \eps$ for those
$ n$ iterates, then specification requires that for every
$ (x_1,n_1),\dots, (x_k,n_k)$ there is $ y\in X$ such that $ y\in B_{n_1}(x_1,\eps)$, then
$ f^{n_1 + \tau}(y)\in B_{n_2}(x_2,\eps)$, and in general

Mixing Axiom A systems satisfy specification; this is a consequence of
the mixing property together with the shadowing lemma.

A continuous potential $ \ph\colon X\to \RR$ satisfies the Bowen property
if there is $ K\in \RR$ such that $ |S_n\ph(x) - S_n\ph(y)| < K$ whenever $ y\in B_n(x,\eps)$,
where $ S_n\ph(x) = \sum_{j=0}^{n-1} \ph(f^jx)$. The following theorem summarizes the classical results
due to Bowen on systems with specification [Bowen1974]. In fact, Bowen required
the slightly stronger property that the shadowing point $ y$ in
(5.2)
be periodic, but this is only necessary for the part of his results dealing with
periodic orbits, which we omit here. ×32

Theorem 5.1.

If $ (X,f)$ is an expansive system with
specification and $ \ph$ is a potential with the Bowen property, then
there is a unique equilibrium measure $ \mu$. This includes the case
when $ f|\Lambda$ is topologically mixing and uniformly hyperbolic, and
$ \ph$ is Hölder continuous.

5.2. Non-uniform Expansivity and
Specification

Various weaker versions of the specification property have been
introduced in the literature. The one which is most relevant for our purposes
first appeared in [Climenhaga and Thompson2012] for MMEs in the
symbolic setting, and was developed in [Climenhaga and Thompson2013], [Climenhaga and Thompson2014], [Climenhaga and Thompson2016] to a version that
applies to smooth maps and flows.

Given $ \eps> 0$, consider the
'non-expansive set'
$ \NE(\eps) = \{x\in X \mid \Gamma_\eps(x) \neq \{x\}\}$, where $ \Gamma_\eps(x)$ is as in
(5.1)
. Note that $ (X,f)$ is expansive if and only if $ \NE(\eps)=\emptyset$. The
pressure of obstructions to expansivity
is The idea of ignoring measures sitting on $ \NE(\eps)$ was introduced
earlier by [Buzzi and Fisher2013]. ×33

In particular, expansive systems have $ \Pexp(\ph)=-\infty$. It follows from
the results in [Climenhaga and Thompson2016] that the
condition $ \Pexp(\ph) < P(\ph)$ is enough for existence of an equilibrium measure.
For uniqueness, we need to weaken the notion of specification. The idea
behind this is to only require the specification property
(5.2)
to hold for a certain 'good' collection of orbit
segments
$ \GGG \subset X\times \NN$ (and similarly for the Bowen property). One must also require
$ \GGG$ to be large enough, which in this case means that there are
collections of orbit segments $ \PPP,\SSS\subset X\times \NN$ that have small pressure
compared to the whole system, but are sufficient to generate $ X\times \NN$
from $ \GGG$ by adding prefixes from $ \PPP$ and suffixes from
$ \SSS$. One can sum up the situation by saying that "the
pressure of obstructions to specification is small". A related idea of studying
shift spaces for which "the entropy of constraints is small" appeared in [Buzzi2005], where Buzzi studied shifts of quasi-finite type
. For more details on the relationship between the two notions, see [Climenhaga2015], especially Theorem 1.4. ×34

We describe two examples for which Theorem 5.2 applies. One
of them is the Mañé example [Mañé1978], which was introduced
as an example of a robustly transitive diffeomorphism that is not Anosov.
This "derived from Anosov" example is obtained by taking a 3-dimensional
hyperbolic toral automorphism with one unstable direction and performing a
pitchfork bifurcation in $ E^{cs}$ near the fixed point so that
$ E^c$ becomes weakly expanding in that neighborhood. One obtains
a partially hyperbolic diffeomorphism with a splitting $ E^s\oplus E^c\oplus E^u$ such
that $ E^c$ "contracts on average" with respect to the Lebesgue
measure; this falls under the results in [Castro2004] mentioned above, and its inverse map
(for which $ E^c$ "expands on average") is covered by [Alves and Pinheiro2010], [Alves and Li2015].

Now given any Hölder continuous potential $ \ph\colon \mathbb{T}^3\to \RR$, it
is shown in [Climenhaga et al.2015] that there is a
$ C^1$-open class of Mañé examples for which this
potential has a unique equilibrium state. In particular, when $ f$ is
$ C^2$, there is an interval $ (t_0,t_1) \supset [0,1]$ such that the geometric
$ t$-potential $ -t\log|\det(df|E^{cu})|$ has a unique equilibrium state for
every $ t\in (t_0,t_1)$, and $ \ph_1$ is the unique SRB measure.

A related second example is the Bonatti–Viana example
introduced in [Bonatti and Viana2000]. Here one takes a
4-dimensional hyperbolic toral automorphism with $ \dim E^s=\dim E^u=2$, and makes
two perturbations, one in the $ E^s$-direction and another one in the
$ E^u$-direction. The first perturbation creates a pitchfork bifurcation
as above in $ E^s$ and then "mixes up" the two directions in
$ E^s$ so that there is no invariant subbundle of $ E^s$; the
second perturbation does a similar thing to $ E^u$. One obtains a
map with a dominated splitting $ E^{cs}\oplus E^{cu}$ that has no uniformly
hyperbolic subbundles.

The same approach as above works for the Bonatti–Viana
examples, which have a dominated splitting but are not partially hyperbolic;
see [Climenhaga et al.2015]. In this case the presence of
non-uniformity in both the stable and unstable directions makes tower
constructions more difficult, and no Gibbs-Markov-Young structure has been
built for these examples. Earlier results on thermodynamics of these examples
(and the Mañé examples) were given in [Buzzi et al.2012], [Buzzi and Fisher2013], which proved existence of a
unique MME. These results make strong use of the semi-conjugacy between
the examples and the original toral automorphisms, and in particular do not
generalize to equilibrium states corresponding to non-zero potentials.

Finally, the flow version of Theorem 5.2 can be applied
to geodesic flow in nonpositive curvature. Geodesic flow in negative curvature
is one of the classical examples of an Anosov flow [Anosov1969], and in particular it has unique
equilibrium states with strong statistical properties. Although the issue of
decay of correlations is more subtle because it is a flow, not a map; see [Dolgopyat1998], among others. ×35 If $ M$ is a smooth
rank 1 manifold with nonpositive curvature, then its geodesic flow is
non-uniformly hyperbolic. Bernoullicity of the regular component of the
Liouville measure was shown by [Pesin1977]. It was shown by [Knieper1998] that there is a unique measure of
maximal entropy; his proof uses powerful geometric tools and does not seem to
generalize to non-zero potentials. Using non-uniform specification, Knieper's
result can be extended to the geometric $ t$-potential for
$ t\approx 0$, and when $ \dim M = 2$, it works for any $ t\in (-\infty,1)$,
showing that the pressure function is differentiable on this interval and we
recover the same picture as for Manneville–Pomeau [Burns et al.2016].

In each of the above examples, the basic idea is as follows: one
identifies a "bad set" $ B\subset X$ with the properties that

(1) the system has
uniformly hyperbolic properties outside of $ B$;

(2) trajectories spending all (or
almost all) of their time in $ B$ carry small pressure relative to the
whole system.

For the Mañé and Bonatti–Viana examples,
$ B$ is the neighborhood where the perturbation is carried out; for
the geodesic flow, $ B$ is a small neighborhood of the singular set.

Given an orbit segment $ (x,n)$, let $ G(x,n) = \frac 1n\#\{0\leq k< n \mid f^kx \notin B\}$ be the
proportion of time that the orbit segment spends in the "good" part of the
system. For flows one should make the obvious modifications,
replacing $ \NN$ by $ [0,\infty)$ and cardinality with Lebesgue
measure. ×36 A decomposition of the space of
orbit segments $ X\times \NN$ is obtained by fixing a threshold $ \gamma> 0$
and taking \begin{align*} \PPP &= \SSS = \{(x,n) \mid G(x,n) < \gamma\}, \\ \GGG &= \{(x,n) \mid G(x,k) \geq \gamma, G(f^kx,n-k)\geq \gamma \text{ for all } 0\leq k \leq n \}. \end{align*} Indeed, given any $ (x,n)\in X\times \NN$, one can take
$ p$ and $ s$ to be maximal such that $ (x,p) \in \PPP$
and $ (f^{n-s}x,s)\in \SSS$, and use additivity of $ G$ along orbit segments
to argue that $ (f^px,n-p-s)\in \GGG$, which yields a decomposition $ X\times \NN = \PPP\GGG\SSS$.
Then one makes the following arguments to apply Theorem 5.2.

Assumption (1) above leads to hyperbolic estimates along
trajectories in $ \GGG$, which can be used to prove specification for
$ \GGG$ (condition ( I
) in Theorem 5.2) and the Bowen property on $ \GGG$
for Hölder continuous potentials (condition ( II
)).

The expansion estimates along $ \GGG$ and the
pressure estimates on $ \PPP$ and $ \SSS$ also yield
$ \Pexp(\ph)< P(\ph)$.

This approach establishes existence and uniqueness, and yields some statistical
properties such as large deviations estimates. However, it does not yet give
stronger statistical results such as a rate of decay of correlations, or the CLT.
In the setting when $ X$ is a shift space with non-uniform
specification, results along these lines have recently been established [Climenhaga2015] by using conditions ( I
)–( III
) (or closely related ones) to build a tower with exponential tails, but it is not
yet clear how this result extends to the smooth setting.

6. The Geometric Approach

6.1. Geometric Construction of
SRB Measures.

6.1.1. Idea of
Construction.

Having discussed constructions of SRB and equilibrium measures via
Markov dynamics (SFTs and Young towers) and via coarse-graining (expansivity
and specification), we turn our attention now to a third approach, which is in
some sense more natural and more simple-minded. The first two approaches
addressed not just existence but also questions of uniqueness and statistical
properties; the price to be paid for these stronger results is that the construction
of a tower (or even the verification of non-uniform specification) may be difficult
in many examples. The approach that we now describe is best suited to prove
existence, rather than uniqueness or statistical properties, but has the advantage
that it seems easier to verify.

We start by discussing SRB measure, which for dissipative systems
plays the role of Lebesgue measure in conservative systems and is the most
natural measure from the physical point of view. So in trying to find an SRB
measure, it is natural to start with Lebesgue measure itself; while it may not
be invariant, we will follow the standard Bogolubov–Krylov procedure
of taking a non-invariant measure $ m$, average it under the
dynamics to produce the sequence

and pass to a weak*-convergent subsequence $ \mu_{n_j} \to \mu$; then
$ \mu$ is $ f$-invariant. If we do this starting with
Lebesgue measure as our reference measure $ m$, then it is
reasonable to expect that the limiting measure will have something to do with
Lebesgue, and may even be an SRB measure. In general, though, the
measure $ \mu$ may be quite trivial—just consider the point
mass at an attracting fixed point. ×37

At an intuitive level, this approach is consistent with Viana's
conjecture ([Viana1998]) that nonzero Lyapunov exponents
imply existence of an SRB, since this should be exactly the setting in which
the iterates of Lebesgue spread out along the unstable manifolds and converge
in average to a measure that is absolutely continuous in the unstable direction.
Now we describe how it can be made precise.

6.1.2. Uniform Geometry: Uniform and Partial
Hyperbolicity

In the uniformly hyperbolic setting, this approach can be carried out
as follows. Let $ \RRR$ be the set of all
standard pairs
$ (W,\rho)$, where $ W$ is a small piece of unstable manifold
and $ \rho\colon W\to (0,\infty)$ is integrable with respect to $ m_W$, the leaf
volume on $ W$. Let $ \Mac$ be the set of all (not necessarily
invariant) probability measures $ \mu$ on the manifold $ M$
that can be expressed as

for some measure $ \zeta$ on $ \RRR$; in other words,
$ \mu$ admits a decomposition (in the sense of Fubini's theorem)
along unstable local manifolds, in which conditional measures are leaf-volumes.
Then one can show that $ \Mac\cap\MMM(f)$ is precisely the set of SRB measures
for $ f$. Moreover, Lebesgue measure $ m$ is in
$ \Mac$ and thus since images of unstable manifolds can be
decomposed into small pieces of unstable manifolds, we have $ f_*^km\in\Mac$
for all $ k\in \NN$, so the averaged measures given by
(6.1)
are in $ \Mac$ as well.

In order to pass to the limit and obtain $ \mu\in\Mac$ one needs a
little more control. Fixing $ K> 0$, let $ \RRR_K$ be the set of all
standard pairs $ (W,\rho)$ such that $ W$ has size at least
$ 1/K$, and $ \rho\colon W\to [1/K,K]$ is Hölder continuous with constant
$ K$. Then defining $ \Mac_K$ using
(6.2)
with $ \RRR_K$ in place of $ \RRR$, one can show that
$ \Mac_K$ is weak* compact and is $ f_*$-invariant for large
enough $ K$. This is basically a consequence of the
Arzelà–Ascoli theorem and the fact that $ f$
uniformly expands unstable manifolds; in particular it relies strongly on the
uniform hyperbolicity assumption. Then $ \mu_n\in \Mac_K$ for all $ n$
by invariance, and by compactness, $ \mu=\lim \mu_{n_j}\in\Mac_K\cap\MMM(f)$ is an SRB measure. Thus
we have the following statement.

Theorem 6.1.

Let $ \Lambda$ be a hyperbolic attractor for
$ f$ and assume that $ f|\Lambda$ is topologically transitive. If
the reference measure $ m$ is the restriction of the Lebesgue
measure to a neighborhood of $ \Lambda$, then the sequence of measures
(6.1)
converges and the limit measure is the unique SRB measure for
$ f$.

Now consider the setting where
$ f$ is partially hyperbolic, i.e., for every point $ x$ the
tangent space splits $ T_xM=E^s(x)\oplus E^c(x)\oplus E^u(x)$ into stable, central, and unstable
subspaces respectively with uniform contraction along $ E^s(x)$, uniform
expansion along $ E^u(x)$, and possible contractions and/or expansions
along $ E^c(x)$ with rates which are weaker than the corresponding
rates along $ E^s(x)$ and $ E^u(x)$.

In the situation where the centre-unstable direction $ E^{cu}$ is
only non-uniformly expanding more care must be taken with the above
approach because $ \Mac_K$ may no longer be $ f_*$-invariant:
even if $ W$ is a "sufficiently large" local unstable manifold, its
image $ f(W)$ may be smaller than $ 1/K$, and similarly the
Hölder constant of the density $ \rho$ can get worse under the
action of $ f_*$ if $ W$ is contracted by $ f$.

The solution is to use hyperbolic times
, which were introduced by [Alves2000]. Roughly speaking, a time
$ n$ is hyperbolic for a point $ x$ if $ df^k|E^u(f^{n-k}x)$ is
uniformly expanding for every $ 0\leq k\leq n$. If $ W$ is a local
unstable manifold around $ x$ and $ n$ is a hyperbolic
time for $ x$, then $ f^n(W)$ contains a large neighborhood of
$ f^n(x)$, and the density $ \rho$ behaves well under
$ f_*^n$. Thus from the point of view of the construction above, the
key property of hyperbolic times is that if $ H_n$ is the set of all
points $ x$ for which $ n$ is a hyperbolic time, then the
measures

are all contained in $ \Mac_K$ (after rescaling to obtain a
probability measure). As long as $ \nu_n \not\to 0$, one concludes that
$ \mu=\lim\nu_{n_k}\in\MMM(f)$ has some ergodic component in $ \Mac_K$, which must be
an SRB measure. To get the lower bound on the total weight of
$ \nu_n$, one needs a lower bound on $ \frac1n\sum_{k=0}^{n-1}m(H_k)$, which can be
obtained using Pliss' lemma as long as a positive Lebesgue measure set of
points have positive Lyapunov exponents along $ E^{cu}$.

One can also construct the SRB measure by beginning "within the
attractor": instead of using Lebesgue measure on $ M$ as the
starting point for the sequence
(6.1)
, one can let $ m^u$ be leaf volume along a local unstable manifold
and then consider the sequence

If the attractor $ \Lambda$ is uniformly hyperbolic, the sequence
of measures 6.4 converges and the limit measure is a unique
SRB measure for $ f$. In the partially hyperbolic setting, it was
shown by [Pesin and Sinai1983] that every limit measure
$ \nu$ of the sequence of measures 6.4 is a
$ u$ -measure
on $ \Lambda$: that is, the conditional measures it generates on local
unstable manifolds are absolutely continuous with respect to the leaf-volume
on these manifolds. What prevents $ \nu$ from being an SRB
measure in general is that the Lyapunov exponents in the central direction can
be positive or zero.

Several results are available that establish existence (and in some
cases uniqueness) of SRB measures under some additional requirements on the
action of the system along the central direction $ E^c$ or
central-unstable direction $ E^{cu}$. For example the case of systems
with mostly contracting central directions was carried out in [Bonatti and Viana2000], [Burns et al.2008] and with mostly expanding
central directions in [Alves et al.2000]. A more general case of systems
whose central direction is weakly expanding was studied in [Alves et al.2014].

In these settings one at least has a dominated splitting, which gives
the system various uniform geometric properties, even if the dynamics is
non-uniform. To extend this approach to settings where the geometry is
non-uniform (no dominated splitting, stable and unstable directions vary
discontinuously) some new tools are needed. An important observation (which
holds in the uniform case as well) is that for many purposes we can replace
$ V^u(x)$ itself with a local manifold passing through $ x$
that is $ C^1$-close to $ V^u(x)$. Such a manifold is called
admissible
, and in the next section will develop the machinery of standard pairs, the
class of measures $ \Mac$, and the sequences of measures
(6.4)
using admissible manifolds in place of unstable manifolds.

6.1.3. Non-uniform Geometry: Effective
Hyperbolicity

The difficulties encountered in the geometrically non-uniform setting
can be overcome by the machinery of 'effective hyperbolicity' from [Climenhaga and Pesin2016], [Climenhaga et al.2016]. This approach has the
advantage that the requirements on the system appear weaker, and much
closer to the Viana conjecture. The drawback of this approach is that it is
currently out of reach to use it to establish exponential (or even polynomial)
decay of correlations and the CLT.

Let $ U$ be a neighborhood of the attractor
$ \Lambda$ for a $ C^{1+\epsilon}$ diffeomorphism, and consider a forward
invariant set $ S\subset U$ on which there are two measurable cone families
$ K^s(x)=K^s(x,E^s(x),\theta)$ and $ K^u(x)=K^u(x,E^s(x),\theta)$ that are

thus $ \lambda(x)> 0$ whenever the system "behaves hyperbolically" at
$ x$, while $ \lambda(x)\leq 0$ when one of the following happens:

some stable vectors expand (so $ -\lambda^s(x)< 0$); or

some unstable vectors contract (so $ \lambda^u(x)< 0$); or

the defect from domination is greater than the expansion in
the unstable cone (so $ \lambda^u(x)-\Delta(x) < 0$).

Let $ \alpha(x)$ be the angle between the cones $ K^s(x)$ and
$ K^u(x)$, and given a threshold $ \ba> 0$, let \begin{eqnarray*} \rho_{\ba}(x) := \ulim_{n\to\infty} \frac 1n \#\{0\leq k< n \mid \alpha(f^kx) < \ba\} \end{eqnarray*} be the
asymptotic upper bound on how often the angle drops below that threshold.
Notice that in the case of a dominated splitting, $ \alpha(x)$ is uniformly
bounded away from 0, so there is $ \ba> 0$ with $ \rho_{\ba}(x) = 0$ for every
$ x$; however, for a system with non-uniform geometry it may be
the case that every $ \ba> 0$ has points with $ \rho_{\ba}(x) > 0$.

With the above notions in mind, we consider the following set of
points: \begin{eqnarray*} S' = \left\{x\in S \mid \llim_{n\to\infty} \frac 1n \sum_{k=0}^{n-1} \lambda(f^kx) > 0 \text{ and } \lim_{\ba\to 0} \rho_{\ba}(x) = 0 \right\}. \end{eqnarray*} Thus $ S'$ contains points for which the
average asymptotic rate of effective hyperbolicity is positive, and for which the
asymptotic frequency with which the angle between the cones degenerates can
be made arbitrarily small. Then we have the following result, which is a step
towards Viana's conjecture.

Theorem 6.2.

([Climenhaga et al.2016]) If $ S'$ has
positive volume then $ f$ has an SRB measure.

6.1.4. Idea of Proof

The construction of an SRB measure in the setting of Theorem
6.2
follows the same averaging idea as in Sects. 6.1.1, 6.1.2: if
$ \mu_n$ is the sequence of measures given by
(6.1)
, then one wish to show that a uniformly large part of $ \mu_n$ lies in
the set of "uniformly absolutely continuous" measures $ \Mac_K$.

In this more general setting the definition of $ \Mac_K$ is
significantly more involved. Broadly speaking, in the definition of
$ \RRR$ we must replace unstable manifolds $ W$ with
admissible manifolds
; an admissible manifold $ W$ through a point $ x\in S$ is a
smooth submanifold such that $ T_xW \subset K^u(x)$ and $ W$ is the graph
of a function $ \psi\colon B^u(r)\subset E^u(x) \to E^s(x)$, such that $ D\psi$ is uniformly bounded
and is uniformly Hölder continuous. The Hölder constant for
$ D\psi$ can be thought of as the "curvature" of $ W$.

When the geometry is uniform as in the previous setting, the image
of an admissible manifold $ W$ is itself admissible; this is essentially
the classical Hadamard–Perron theorem. In the more general case this
is no longer true; although $ f^n(W)$ contains an admissible manifold, its
size and curvature may vary with time $ n$, with the size becoming
arbitrarily small and the curvature arbitrarily large. In this setting a version
of the Hadamard–Perron theorem was proved in [Climenhaga and Pesin2016] that gives good bounds
on $ f^n(W)$ when $ n$ is an
effective hyperbolic time
for $ x\in W$; that is, when \begin{eqnarray*} \sum_{j=k}^{n-1}\lambda(f^jx)\ge\chi(n-k) \end{eqnarray*} for every $ 0\le k < n$, where
$ \chi> 0$ is a fixed rate of effective
hyperbolicity
.

The set of effective hyperbolic times is a subset of the set of
hyperbolic times; the extra conditions in the definition of effective hyperbolic
time guarantee that we can control the dynamics of $ f$ on the
manifold itself, not just the dynamics of $ df$ on the tangent
bundle. In the uniform geometry setting from earlier, this extension came for
free for hyperbolic times.

With the notion of effective hyperbolic times, the approach outlined
in Sects. 6.1.1, 6.1.2 can
be carried out. One must add some more conditions to the collection
$ \RRR$; most notably, one must fix $ n\in \NN$ and then consider
only admissible manifolds $ W$ for which \begin{eqnarray*} d(f^{-k}(x),f^{-k}(y))\le Ce^{-\chi k}d(x,y)\quad \text{ for all} \quad 0\le k\le n \quad \text{ and} \quad x,y\in W, \end{eqnarray*} and then
define $ \Mac_{K,n}$ using only this class of admissible manifolds. In addition
to size of $ W$ and regularity of $ \rho$, the constant
$ K$ must also be chosen to govern the curvature of $ W$,
but we omit details here. The point is that the set $ \Mac_{K,n}$ is compact,
but not $ f_*$-invariant, and so the proof of Theorem 6.2 can be
completed via the following steps.

(1) Writing
$ H_n$ for the set of points with $ n$ as an effective
hyperbolic time, use Pliss' lemma and the assumption that $ S'$ has
positive volume to show that $ H_n$ has positive Lebesgue measure on
average.

(2) Use
the effective Hadamard–Perron theorem from [Climenhaga and Pesin2016] to show that
$ \nu_n:=\frac 1n\sum_{k=0}^{n-1}f_*^k\in\Mac_{K,n}$, and use the bound from the previous step to get a lower
bound on the total weight of $ \nu_n$.

(3) Write $ \mu_n=\nu_n+\zeta_n$ and argue from
general principles that if $ \mu_{n_k}\to\mu$, then $ \mu$ has an ergodic
component in $ \Mac$; moreover, this ergodic component is hyperbolic
and $ f$-invariant, so it is an SRB measure.

6.2. Constructing Equilibrium
Measures.

A natural next step is to extend the above procedure to study general
equilibrium states, and not just SRB measures. The direct analogue of the
previous section has not yet been fully developed, and we describe instead a
related approach that is also based on studying how densities transform under
the dynamics.

First consider the case of a piecewise expanding interval map, and the
question of finding an SRB measure. In this case there is no stable direction,
and so we do not have to keep track of the "shape" of unstable manifolds, or
admissible manifolds; indeed, a local unstable manifold is just a small piece of
the interval, and an SRB measure is just an invariant measure that is absolutely
continuous with respect to Lebesgue. Thus the entire problem is reduced to
the following question: given a (not necessarily invariant) absolutely continuous
measure $ \mu\ll m$, how is the density function of its image $ f_*\mu$
related to the density function of $ \mu$? One ends up defining a transfer operator
$ \mathcal{L}$ with the property that if $ d\mu = h\,dx$, then $ d(f_*\mu)=(\mathcal{L} h)\,dx$.
Questions about the existence of an absolutely continuous invariant measure,
and its statistical properties, can be reduced to questions about the transfer
operator $ \LLL$.

The central issue in studying $ \LLL$ is the problem of finding
a Banach space $ \BBB$ (of functions) on which $ \LLL$ acts
"with good spectral properties". Generally speaking this means that 1 is a
simple eigenvalue of $ \LLL$ (so there is a unique fixed point
$ h=\LLL h$, which corresponds to the unique absolutely continuous
invariant measure) and the rest of the spectrum of $ \LLL$ lies inside a
disc of radius $ r< 1$, which guarantees exponential decay of
correlations and other statistical properties.

For piecewise expanding interval maps, this was accomplished by
[Lasota and Yorke1974], and the approach can be
adapted to equilibrium states for other potential functions by considering a
transfer operator that depends on the potential in an appropriate way. A
thorough account of this approach is given in [Baladi2000].

The mechanism that drives this approach is that the expansion of the
dynamics acts to "smooth out" the density function; irregularities in the
function $ h$ are made milder by passing to $ \LLL h$. (The
precise meaning of this statement depends on the particular choice of Banach
space $ \BBB$, and is encoded by the Lasota–Yorke inequality,
which we do not pursue further here.) But this means that one runs into
problems when going from expanding interval maps to hyperbolic
diffeomorphisms, where there is a non-trivial stable direction; the contracting
dynamics in the stable direction make irregularities in the function worse!

In the classical approach to uniformly hyperbolic systems, this was
dealt with by passing to a symbolic coding by an SFT (as described after
Theorem 1.1) and then replacing the two-sided SFT
$ \Sigma \subset A^\ZZ$ by its one-sided version $ \Sigma^+ \subset A^\NN$. As described after
Theorem 1.3, the transfer operator $ \LLL$ has an
eigenfunction $ h\in C(\Sigma^+)$, and its dual $ \LLL^*$ has an eigenmeasure
$ \nu \in C(\Sigma^+)^*$; combining them gives the equilibrium state $ d\mu = h\,d\nu$.
Note that positive indices of an element of $ \Sigma$ code the future of a
trajectory, while negative indices code the past, and so dynamically, passing
from $ \Sigma$ to $ \Sigma^+$ can be interpreted as "forgetting the
past". Geometrically, this means that we conflate points lying on the same
local stable manifold; taking a quotient in the stable direction eliminates the
problem described in the previous paragraph, where contraction in the stable
direction exacerbates irregularities in the density function.

More recent work has shown that this problem can be addressed
without the use of symbolic dynamics. The key is to consider a Banach space
$ \BBB$ whose elements are not functions, but are rather objects that
behave like functions in the unstable direction, and like distributions
in the stable direction. For SRB measures, this was carried out in [Blank et al.2002], [Gouëzel and Liverani2006], [Baladi and Tsujii2007]. A further generalization to
equilibrium states for other potential functions was given in [Gui and Li2008]; as with expanding interval maps,
this requires working with a transfer operator $ \LLL$ that depends on
the potential. Moreover, instead of distributions along the stable direction,
one must consider a certain class of "generalized differential forms". We refer
the reader to ([Gui and Li2008], Sect. 7) for a comparison of this
approach to equilibrium states and other related approaches, including the
technique of "standard pairs".

It remains an open problem to extend this approach to the
non-uniformly hyperbolic setting.

6.3. Ergodic Properties

An important open question is to study uniqueness and statistical
properties of the SRB measure produced in Theorem 6.2, or of any
equilibrium states that may be produced by an analogous result for other
potentials. One potential approach is to study the standard pairs
$ (W,\rho)$ and derive statistical properties via coupling
techniques, as was done by Chernov and Dolgopyat in another setting ([Chernov and Dolgopyat2009]). Coupling
techniques are also at the heart of Young's tower results for subexponential
mixing rates ([Young1999]). ×38 One might also hope to adapt the
functional analytic approach from Sect. 6.2 into
the non-uniformly hyperbolic setting and obtain statistical properties this way.
For now, though, we only mention results on Bernoullicity and hyperbolic
product structure.

6.3.1. SRB Measures

By a result of [Ledrappier1984], a hyperbolic SRB measure has at
most countably many ergodic components and every hyperbolic SRB measure
is Bernoulli up to a finite period. It follows that there may exist at most
countably many ergodic SRB measures on $ \Lambda$. One way to ensure
uniqueness of SRB measures is to show that its every ergodic component is
open (mod 0) in the topology of $ \Lambda$ and that $ f|\Lambda$ is
topologically transitive.

6.3.2. Equilibrium Measures

Let $ \mu$ be a hyperbolic ergodic measure for a
$ C^{1+\alpha}$ diffeomorphism $ f$. Given $ \ell> 0$,
consider the regular set $ \Gamma_\ell$, which consists of points
$ x\in \Gamma$ whose local stable $ V^s(x)$ and unstable $ V^u(x)$
manifolds have size at least $ 1/\ell$. For $ x\in \Gamma_\ell$ and some
sufficiently small $ r> 0$ let $ R_\ell(x,r)=\bigcup_{y\in A^u(x)}V^s(y)$ be a rectangle
at $ x$, where $ A^u(x)$ is the set of points of intersection of
$ V^u(x)$ with local stable manifolds $ V^s(z)$ for $ z\in\Gamma_\ell\cap B(x,r)$.
We denote by

Say that $ \mu$ has a direct product
structure
if the holonomy map is absolutely continuous with the Jacobian uniformly
bounded away from 0 and $ \infty$ on $ R_\ell(x,r)$.

Conjecture 6.3.

If $ \mu$ is a hyperbolic ergodic
equilibrium measure for the geometric $ t$-potential for a
$ C^{1+\alpha}$ diffeomorphism $ f$, then $ \mu$ has a
direct product structure.

If true, this would imply that $ \mu$ has some "nice" ergodic
properties; for example, it has at most countably many ergodic components.
Similar results have recently been established (using the symbolic approach)
for two-dimensional diffeomorphisms and three-dimensional flows ([Sarig2011], [Ledrappier et al.2016]).

We conclude with a conjecture on the relationship between effective
hyperbolicity (from Sect. 6.1) and decay of correlations. Suppose that
$ \Lambda$ is an attractor with trapping region $ U$, and that
we have invariant measurable transverse cone families defined Lebesgue-a.e. in
$ U$, with the property that there is $ \chi> 0$ for which
\begin{eqnarray*} S'=\left\{x\in U\mid\llim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\lambda(f^kx)> \chi\text{ and } \lim_{\ba\to 0}\rho_{\ba}(x)=0\right\} \end{eqnarray*} has full Lebesgue measure in $ U$. Consider for each
$ N\in \NN$ the set \begin{eqnarray*} X_N=\left\{x\in U\mid\sum_{k=0}^{n-1}\lambda(f^kx)> \chi n\text{ for all } n> N\right\}, \end{eqnarray*} and note that the assumption on
$ S'$ guarantees that $ m(U{\setminus} X_N)\to 0$ as $ N\to\infty$.

Conjecture 6.4.

If $ m(U{\setminus} X_N)$ decays exponentially in
$ N$, then the SRB measure $ \mu$ produced by Theorem
6.2 has
exponential decay of correlations.

Some support for this conjecture is provided by the fact that the
analogous result for partially hyperbolic attractors with mostly expanding
central direction was proved in [Alves and Li2015].

Acknowledgements.

We are grateful to the
anonymous referee for a very careful reading and for many thoughtful and
insightful comments that have improved this paper. We are also grateful to
ICERM and the Erwin Schrödinger Institute, where parts of this work
were completed, for their hospitality.

× Katok, A., Hasselblatt, B.:
Introduction to the modern theory of dynamical systems. Encyclopedia of
Mathematics and its Applications, vol. 54, Cambridge University Press,
Cambridge (1995). With a supplementary chapter by Katok and Leonardo
Mendoza